Hello; this is definitely quite strange.
We haven't had any problems in our test labs, using significantly less-powerful hosts and significantly more traffic. Other users aren't reporting this problem on any platform, so I'm inclined to say it's configuration-related, but what configuration?
How is ProGet configured? If it's just a single feed and a single connector to that feed, then it's not your Proget configuration.
In any case, I'm certain it's not the database itself, but it's related to the network stack related. SQL Server uses network connections, and the "connection timeout" happens when the network stack gets overwhelmed. This can happen when a TON of connections are open, but not closed.
One thing we've seen is that certain network-based reporting tools (monitoring/logging) end up causing problems. They try to send a error over the network that the stack is overloaded, which then gets queued up, and continues to overload the stack. Eventually it calms down.
We've also seen bad hardware cause this. One time, it was even a bad wireless access point in. No idea how that happened, but something to do with routing and packets.
So perhaps try a new server, like make one at LightSail , totally fresh. If you can find a way to reproduce it, then it'll be good, because we can at least investigate it further then.