When I start evolution, I get a message about TLS negotiations with the mail
server failing (actual message to follow), and am prompted to enter my password
for each account. The passwords are already pre-filled. Clicking OK or Cancel
does the same thing, i.e. it cycles through the list of password dialogs.

pkill evo is the only answer. Restarting evolution makes the problem go away.

Almost definitely related to bug 215634.

evolution-2.10.2-3.fc7
evolution-data-server-1.10.2-3.fc7