The key to this is method of troubleshooting. There is no button that you can push that says 'tell me the exact problem I'm having in the environment right now', so you have to work your way through the issue.
Given you've said that the appliance does function properly most of the time, I'm not sure I'd start there. While it's possible that the appliance has a problem, it isn't a universal problem, so just trying to comb through appliance logs will likely be unfruitful - you want to look through them with a finer sift than that. I do like the angle Marianne suggests with 100031616, as there is some chance you're exhausting a resource and recovering, but that's a 50/50 to me.
You indicate there isn't a pattern. I have that feeling a lot when I'm dealing with a complex problem and I'm almost always proven wrong. There's a pattern, it's just not yet visible. Start there. If you've got failures going back in time, go back as far as you can and start isolating the clients that are having the issue. Look for a pattern of client, physical network, OS, time of backup, policy type, etc. If you're seeing a resource exhaustion, you may see a time-based pattern, i.e., when you have a lot of database or VM backups all starting at one time you start to see these failures.
I tend to write these things down or build a spreadsheet so I can more easily sort the data. Start trying to answer questions like
- How many failures have I had across the search period?
- Which clients?
- Do these clients always fail or do they get successful backups periodically?
- When failures occur, is it one client or multiple clients in the same period of time?
- What is different about the successful vs unsuccessful backups (time window, SLP structure, other jobs running, etc.)?
- Is replication running during successful windows?
- Is replication running during unsuccessful windows?
- Are the failures always on the same day?
- Do the failures occur during other datacenter activities?
If there's no discernible pattern in that data, I'd then look at which client(s) have experienced this the most in the last 30 days. For starters, look at the job info in the bptm and bpdm logs for all of the jobs that have failed, and correlate the time of the failures to system logs on the appliance, looking for something to show up. If you find a given failure, do the same thing for another client at another time to see if the same failure or type of failure occurs on the system.
Sorry if this seems rudimentary, but like I said there's no button to push or switch to throw that's necessarily going to tell you exactly what is wrong. This is just how I'd start to do the footwork of figuring out what it is that's going on because it isn't immediately obvious.