Sort of, but
Someone could do this, but most states ban it either explicitly or implicitly. In some states it is implicitly blocked by requiring that someone be registered in that party. Since you can only register in one party at a time, you can't vote in both party primaries if they're held at the same time.
In Nevada, someone noticed that due to differing rules on party registration for the two major parties, it was possible to be registered as a Republican until after the deadline and then use same day registration to vote in the Democrat caucus. It is explicitly against the law to vote in both caucuses, which Nevada announced that it was explicitly enforcing. So it is certainly possible to vote in both party caucuses in Nevada, but it is illegal.
It also may be possible to vote in primaries in different states if you change your residence. Note that presidential primaries range from February to June.