Back in the late 70s and early 80s, US Department of Defense was readily preparing for USSR invasion of Iran, which turned out to be a complete blunder in intelligence. Are we experiencing the same thing with warnings of possible Russian invasion of Baltic states? I'm intrigued whether this is just Western propaganda, or are there clear indications, and strategic risk that Russia might just do that?
Anxiety about the Baltic states can be best understood by what's happening in Ukraine. The eastern part has been annexed by rebels, who quickly created the Donetsk People's Republic. The main point of the Donetsk People's Republic, much like Crimea, Abkhazia and South Ossetia, is their unification with Russia. Sympathy with Russia is widespread in these regions as many identify as Russian, and in the DPR's case, feel themselves so Russian they have to go around changing the wording on postboxes by one letter just to ensure that the word is Russian and not Ukrainian.
It has been argued that these are examples of a strategy of frozen conflicts by Putin to leverage control over his neighbours, likely because of fears of being unable to defend against NATO encirclement. Lithuanian officials are concerned. Their foreign minister told the BBC that "Russia is not a super power, is a super problem", while a senior Lithuanian military officer said: "It is an alarm that says not that they are coming but that the threat is growing and is growing every day." Evidently they feel the risk is real.
This is important to bear in mind for two reasons. Firstly, because there are one million ethnic Russians across the Baltic states. Secondly, because Putin has expressed a desire to defend Russian speaking citizens. Many worry "defend" is a synonym for "unite". Indeed after this statement was made The Economist decided to mock the idea, and created a map of the world operating under Putin's implicit logic of "Linguistic Imperialism".
To be fair, I have some qualms about the possibility of answering a question of this kind as it is in essence un-assessable. But, let's give it a try... Also, before reading further, I'd like to say that as I'm most familiar with Estonian media, I will use that to exemplify the rest of the Baltic states.
This is an interesting topic and one in which your point of view matters a lot. I know people in the Baltic states (mostly Estonia as above), Finland, Western Europe, the USA and Russia, and how any of these people treat a question of this magnitude is down to the local media and their own understanding. Several medias, I find, are too prone to highlight danger and to create public hysteria (the more hyperbolic a headline, the more links clicked; the more links clicked, the more ad revenue). However...
I would firstly emphasise that both Russia will have military plans in preparation for invading the Baltic countries and all of the Baltic countries have readily available defensive plans. These in itself are not an indication that the threat is immediate, but rather an indication that the countries' respective military organisations are fulfilling their purpose. I would also like to downplay the importance of the Baltic countries' militaries noting that they are in danger, as a) they are biased, and b) they benefit from it. Similar swashbuckling on the other side is only to be expected.
This paragraph will be hyperbole in full, but I figure it can indicate the limit of what we know which is incredibly relevant for this question. The Russian abduction of an Estonian security agent has been noted in the other answers. But, again, it is impossible to know whether that was due to internal FSB politics, an order from someone in the Russian Government, or something entirely different. Once the act had been committed, the Russian side would have lost face by backing down and that would come with repercussions in domestic politics. This also means that a few warmongering individuals could cause a lot of trouble...
But, back to more fact-based reasoning:
The Russian-speaking population of the Baltics has been noted in the other answers, and in general they are right, these people provide the casus belli that Russia has used in other areas over the last few decades (@inappropriateCode referenced this well in his answer, suggest you look there). Even if the vast majority of these people in the Baltic states are happy living where they are, I would be uncertain about the methodology of any such polling; I am also sure that if the Russian propaganda divison so wanted, they could draw up similarly exhaustive polls showing near enough full support for 're-unification'. The true feelings of the people will be quite difficult to find out, however, including the conditions on which these are dependent.
It is not unimportant that many Russians in the Baltic states can easily feel as second-class citizens (the article references older people for whom the problem is likely to be more real). Indeed, only the present government of Estonia in that country has shown some willingness (and even that staunchly opposed by the Opposition) (also, sorry, this link is in Estonian as I couldn't find an English equivalent) to deal with the grey passports. Similarly, the use of Russian as a language of communication in certain regions of the countries disadvantages the people from there with regards to opportunities elsewhere in the land. However, this can be opposed by the fact that these people also generally have the opportunity to travel and work within the European Union where they can get by with English, and can probably have an easier time establishing a higher standard of living than in their home country (or than they could in Russia). Hence, even this citizenship comes with a considerably wider range of options for the people than a Russian one would (although I think the Telegraph was stressing the opposite here, in a typical scare-mongering piece), reducing the wish for a Russian takeover where such might otherwise exist. Mind, the Russian government would be "listening" to the side whose point of view they like...
What else should be considered after having established that a suitable military plan exists and that a casus belli is manufacturable?
I would start with the question of who can order such an attack. The Russian constitution specifies that the Federal Assembly is in charge of using armed forces outside of the country, however, it would not be a stretch of imagination to say that at present, President Putin's influence in the country dwarfs such considerations, and essentially he rules as an autocrat but with a potentially weaker hand than in the previous decades.
Hence, the next question: What would prompt President Putin to determine that it would be in the country's best interest to invade a Baltic state despite considerations of NATO protection?
It is accepted that all countries need external enemies to draw attention away from domestic problems. This, indeed, would be the main point of my answer: if the domestic situation of the Russian Federation deteriorates or some sort of internal threat to the regime surfaces, the risk that Russia will strike out at foreign powers (including the Baltic states) increases. However, I would also add that Russia can probably find easier targets as the Baltic states are generally do well (Estonia has been at the target for a while) with regards to the suggested 2% defense spending which Mr Trump has made the condition of support in the alliance during his term.
To sum up: There is some risk, but the above points indicate that easier gains elsewhere are possible if an external enemy needs to be manufactured by the Russian state (if domestic situation becomes worse for the ruling elite). However, the truth is we really do not know as there is no fixed criteria this is dependent on, and we really cannot know.
There is no risk because there is no desire of the ethnic Russians in the Baltic states to live under Russian rule. The problems with the ethnic Russians there are the opposite of the situation in Ukraine and previously in South Ossetia and Abkhazia that led to military conflict. In the Baltic states, the ethnic Russians want to be full citizens of the country they live in, but they face obstacles due to not speaking the local language which is required by law.
The fear of a Russian intervention in the Baltic states does exist, and it's that fear that Russia is exploiting. From Russia's point of view, it's not all that bad that NATO is considering beefing up the defense of the Baltic states precisely because Russia is not interested in doing anything militarily there. This means that Russia is going to do its best to look more threatening, like stationing the S-400 system and Iskander missiles in Kaliningrad. Also Russia is targeting the ethnic Russians with anti-Western propaganda.