(I'm marking this as the answer to my question in the absence of more info, but feel free to add other answers if you have more to provide.)
According to http://www.janda.org/ICPP/ICPP2000/Countries/9-CentralEastAfrica/98-Uganda/98-Uganda63-00.htm
On June 29 2000, the referendum was held regarding a return to a
multi-party system or not. Support for the no-party "movement" system
received an overwhelming 90 % of the votes. However, less than 50% of
the electorate turned up to vote at all, and critics claim this is a
sign of no support of the Museveni no-party system. Instead, the
referendum should be seen, especially according to the UPC, as a
rejection of the no-party system. (Keesings, 2000: 43610)
(The UPC is a political party which of course would oppose a no-party system, and I'm not sure that a low turnout would necessarily be a sufficient indication of a lack of support for the system.)
The article continues:
Recent charges of corruption directed at the Museveni government, and
the subsequent resignation of his brother Salim Saleh, has tainted
Museveni's image as a crusader against official corruption. Museveni
will have to convince his opposition and his supporters that first of
all, he still takes a strong stance against corruption, and second of
all, that the no-party system is not a scheme to keep him in power.
There are fears that with Museveni coming out victorious after the
referendum, Uganda will move to become a single-party state. Even
though the peace and stability that Uganda has enjoyed under
Museveni's rule is appreciated, recent developments have left many
people feeling anxious about the future development of Uganda. The
electorate is becoming younger and with that the memories of the Amin
and Obote regimes are fading. To them, a multi-party system no longer
presents itself as destabilizing and destructive.
I'd still like more background, especially since the rejection of the non-party system could not seem to be due solely to dissatisfaction with the president or otherwise he would presumably not have been reelected after the reinstitution of parties, but it does seem from the above that the rejection of the non-party system could be due to:
- Performance of the ruling administration as far as corruption
- Concerns of a non-party system being a tool of the incumbent
- A waning of support due to the aging out of those who had seen harmful effects of factional fighting (with the younger generation perhaps swayed by international norms that "multi-party" equals freedom?)
Given that the system remained in place from 1986 to 2005 despite a chance for replacement as late as 2005, it appears to me that a non-party system is nevertheless tenable under the right circumstances and public support, particularly if the public is concerned about the consequences of political divisiveness.