Typically, a 500 error isn't strictly browser related, but rather server related. We most commonly see the "500 Internal Server error" which is just a generic "something's wrong" type of thing. The cause could be pretty much anything, but is seldom client-side. However, a bad server-side script, or a server-side script reading a bad header or cookie, could potentially lead to the same error.
The more common errors we see, which have the symptoms of multiple logins are due to client-side cookies. These can be handled (and 99% of the time remedied) by clearing your cache (you can Google for specific directions pertaining to your browser of choice). Using a different browser (especially a seldom used one) also works as these cookies are stored specifically per browser, so it acts just like clearing the cache.
We see this situation often, and people always recommend changing browsers, but what we see is that they all recommend different ones

, all of the 4 major have been faulty and the solution, which leads me to believe that it's the cache of each moreso than the browser engines themselves.
So, all of this is my very long winded and overly technical way of saying that if you're having issues, the first thing to do would be to clear your cache, and that should solve most of them.