You could try running this from the Command Line on the server:
fmsadmin STATUS file DatabaseName
I can't test this at the moment to see what it returns but it might indicate the normal/paused status? Obviously this has to be run from the context of the FileMaker Server machine so might not help in the end . . . .
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I don't think you can... why would you need it?
"why would you need it?"
If a database is paused, a connected client can trigger a script that can hang (or at least cause the appearance to hang) the client. If an automatically triggered script could detect a paused state, it could cleaning announce such and exit.
Unless you're doing something to explicitely pause FMS, FMS only pauses the files at the very end of a regular backup schedule. By nature that's very fast and usually not noticeable by the clients. Are you seeing long pauses? And if so can you share some details about your deployment?
The question pertains to databases that have been manually paused using the Admin Console.
Why would that happen? That seems more like a procedural issue, agreement on maintenance windows outside of regular business hours and that sort of thing.
If that is not possible and whoever does this will continue to do this and then you need to raise a red flag because this is a very risky deployment and you can not even begin to guaruantee any sort of uptime. In that case you probably need to set up some sort of monitoring on the FMS event logs (or on Windows on the Application event log) that can send out email as soon as the files are getting paused or closed...