Safe deposit boxes require two keys to open them — the key the bank gives you and the bank's master key. Without both keys the only way to open the box is to drill it open. Not even the bank can open it without drilling it. For a government agency to get the right to search it, and possibly seize the contents, it must have a court order authorizing access. Without your key, the agency would also have to drill it open. When I was a revenue officer for the IRS I had to do that a couple of times. It wasn't a simple or fast thing to accomplish. I made sure I had a pretty idea that there was something valuable enough in there to make the effort worthwhile. I could seal the box easily enough so that no one, including the box holder, could get into it without me being there. But actually getting access to the box without the cooperation of the taxpayer was another matter.
The best use for a safe deposit box is holding the original version of important documents like your passport, birth certificate, etc so they won't get lost or destroyed should some disaster strike your home (e.g. in a fire in your home). The one original documenst you don't want to keep there are is your will. That's because the will is what the court needs to appoint your chosen executor and your executor is the one needed to open the box. Putting the original will (a copy is fine) in the box will slow down probate proceedings and that can result in problems when your assets are left unattended. There are a few other documents I wouldn't suggest go into a safe deposit box, too.
What fear do you have regarding the FDIC or other bank regulator? They aren't interested in what's in your safe deposit box. They are interested in how well the bank is following federal and state banking laws and that the bank's finances are sound. The FDIC in particular is concerned with bank solvency so that it won't have to pay out any more insurance claims than absolutely necessary. When a bank is in financial trouble, the FDIC will examine the books of the bank and if it appears that deposits might be at risk, it will arrange for the troubled bank to be absorbed by one that is sound so that the insured deposits are safe. None of that has anything to do with bank customer safe deposit boxes.