I also expect to get notified of incoming messages - it was not infrequent that I would restart Teams and get 3-5 notifications of messages all at once, timestamped over the previous several hours.

As best I could tell, my client would simply lose connection to the server entirely silently. Normally I would find out when trying to send a message to someone, and noticing the delivery animation/indicator was not appearing. Restart Teams and boom, here's a little pile of messages I'd missed.