No, the stash space is not causing the problem, it's just one of the consequences. It seems well timed, because it is running out of stash space (for your MP replay, which is less important) before the actual connection breaks down (which is more important).
To explain, the packets for the MP replay are stored in a "stash" which is of limited size (specially so it doesn't break anything else). If you have a problem with the network (missing packets from the host) your stash can build up with your own position packets. Eventually the stash space runs out and you get a warning. This is no real problem except some of your car's position packets will not be saved in the replay. Then shortly after, caused by the same problem, you are disconnected from the server, due to lack of any packets from it, or its lack of packets from you.