Hi @jyip_5228,
Just an update! We will be shipping a potential fix in PG-1783, which adds a new checkbox in advanced settings (unchecked by default):
Close Database Connections Early
EXPERIMENTAL (for debugging/testing only) - As of ProGet 5.3, database connections are left open during the lifecycle of a NuGet API request as a means to reduce overhead; however, this may be causing ProGet to run out of available connections to SQL Server. Set this value true to open/close database connections as needed on NuGet feeds.
We'll update when this is shipped --- but if we can get some folks to verify that this works better (we can't repro, at all
), then we will likely make it the default. Hopefully this will do it. Seems better than raising connection pool limits 
If so, then the savings in connection open/close overhead don't seem to make it particularly worthwhile. This "keep open" technique made a ton of differences elsewhere in our software, but since a NuGet API requests may yield a ton of other network requests (via connectors) and block, the pool may be getting drained too quickly...
just a theory, as I mentioned. Anyway hope this helps, stay tuned!!
Thanks!