FYI -- for NuGet feeds, a "Healthy" connector means that the service was able to connect to the feed and query the total number of packages at the source. Other connectors (npm, maven), the entire index is downloaded, so "healthy" means there is an offline index. Unfortunately NuGet does not support an index download.
I have heard a few other reports of behavior like this from when connecting to a Nexus3 repository, but I never heard back regarding a resolution. I do know, for example, that connecting to a TeamCity feed can sometimes be problematic because it doesn't always support API v2. But upgrading to TC v9 and enabling the proper API is a resolution there.
Generally speaking, there are three possibilities:
ProGet is being blocked from making a request to the remote repository, by a firewall or some such; this should be logged
The remote repository is not returning the results; e.g. package not found, when it should be
ProGet is not interpreting the resulst correctly; e.g. package is in the results, but it's not shown to the user
Requests to connectors are not logged; there are simply too many and too much data. But, it's easy to capture them by attaching a proxy server, like Fiddler, between ProGet server and the remote repository. To do this, install Fiddler on the server, then configure ProGet's proxy settings (under Admin) to point to Fiddler. Soon enough, you should see the web requests that ProGet is making.
Once that's configured, you can now identify which scenario it is from above, and in the case of #3, we can certainly try to identify why that's happening.In the case of #1 or #2, it's difficult because it's probably a third-party feed, but we can try to help!