I just received this email from an user that repeatedly complains that JIRA kicks him out after period of inactivity, even if he checks the 'remember me' box on the login screen:

Figured it out... really brain dead on my part.
On another PC I set the remember me checkbox, so only after logging out on that machine did it allow the other machine to keep logged in.

I've had multiple reports of JIRA not remembering logon, but so far this comes closest to reproducible scenario.