My initial thought was that you accidentally used a key value for your cookie, which is reserved by Umbraco, which could lead to an incorrect cookie being read, which would cause problems. The solution to this is to simply rename your cookie.
If this is not the case, I have another theory:
HTTP requests will always include all cookies whose path / domain matches the domain of the requested resource. They are sorted by the length of the path in the first place, and secondly, by the time of creation. If for some reason the Umbraco backend finds that the cookie used to authenticate by its index number (not even surprised) in the list, and not by the key value, your user cookie will cause the index to shift, which causes Umbraco to look at the wrong cookies
So, if renaming the cookie did nothing, an interesting task would be to set the cookie path to the shortest path that will force your browser to place the cookie further down the list, so the index will not move.
This is just a theory, so I'm curious to know how this happens :)