Thank you for your response. I already opened a support ticket with Esri regarding this issue and I was asked to do the same thing. It usually has about 15 connections to the Data Store, but quite awhile ago we changed the maximum number of connections from 150 to 300 since we do utilize the ArcGIS Data Store quite a bit (we have about 350 users). But I've never seen the number of connections anywhere near 150 or 300. Tech support and I are at the point where we need the error to happen again so we can see how many connections are being made to the database to confirm it isn't above the maximum allowed.
With that being said, I did some more investigating this morning and was able to find the previous time this error happened. Both times that this error happened, the CPU usage on the Data Store server (dedicated VM with only Data Store installed) spiked to 100%. It never spikes to 100% any other time, so it probably isn't a coincidence that we're receiving the error message the same time CPU is at 100%. Unfortunately, I can't see what caused the CPU to jump to 100%, but I will look the next time this happens.