Yes, I have seen the QC Server where an ''API Client'' (like the way QC-Sync communicates with QC) can pile up too many connections to the QC Server.
Sometimes it gets ''hung'' and refused future connections. This can be resolved by the QC Admin selecting the connection from the Site Admin tool and clearing it --- or with a cycle of QC services on the QC server.

I saw this when trying to get some VB script code to communicate via API with QC and after some ''cycles'' of my script (when running it many times), it would ''hang'' that connection and future connections attempted from my script until I cycled QC services.

Let's hope this does not happen with your QC-Sync setup going forward - I have never seen it required for QC-Sync before.