I had a user who was having a crashing issue with our app and was not closing his connection to the database properly - eventually reaching his connection limit and was unable to log back in.
That's ok. The problem was eventually fixed and the app is closing the connection properly now.
I was wondering, if a user reaches his connection limit, and without me doing anything, what is the default setting for Postgres 9.1 to drop the connection on its own?
Well, for your actual question:
if a user reaches his connection limit, and without me doing anything, what is the default setting for Postgres 9.1 to drop the connection on its own?
If the connections you are talking about are active connections to real clients (i.e. the client is still around, not crashed and exited, and can read and write to its connection socket), Postgres will not "drop the connection on its own". See this question if you're interested in ways to prune such idle clients.
On the other hand, it is possible for a Postgres backend to be nominally connected to a client which has really crashed or otherwise exited, as you alluded to earlier. In that case, Postgres may not notice the client is gone and not just in an idle state for some time. I believe the exact time would be controlled by a "TCP keepalive time" OS setting, or you could even fiddle with this setting in your client via connection parameters: see the various keepalives-related settings.
