[NF] SQL Server - too many databases?

We are running into an intermittent issue w/ our SQL Server running SQL2016. It's not that our tables are all that large (usually), but our server is trying to manage around 550 different DB's parked on it. Few of the DB's are open at any one time, but the server has to keep track of them at least to some extent.

When the problem arises our dev team runs into timeouts and sometimes the problem goes away, some time it doesn't and our IT guys out of desperation restarts the machine.

One of our devs suggested that even though we aren't past some theoretical limit on the number of DB's the server can manage, maybe we just have too many and getting rid of most, or at least detaching them, might help.

Any thoughts on this?