I did a little bit more research last night, and found out the errors actually start after 15 minutes. Not 10.
I was reading a blog by Scott Guthrie yesterday, which indicated there are differences between the timeout of sessions, and those of authentications. I'm starting to believe this might be the origin of my troubles, but I will have to look into things a bit more.
My girlfriend noticed that the "remember me" option doesn't seem to do too much. Every time she comes back to the site, the site seems to have forgotten she was there. That, in my opinion, does hint at a problem with the cookies. Will try to dig deeper this weekend...