The U.S. Copyright Office says of poor man's copyright:
There is no provision in the copyright law regarding any such type of protection, and it is not a substitute for registration.
This means that the U.S. government will not grant you any special rights or protections for having undertaken poor man's copyright, unlike advance copyright registration, which confers the ability to collect statutory damages. In any case, as a prerequisite to legal action, you will need to register your work, even if you must do so after the infringement has taken place:
You will have to register, however, if you wish to bring a lawsuit for infringement of a U.S. work.
Instead, poor man's copyright is purely an evidentiary mechanism for proving possession of a work at a particular point in time. Suppose a best-selling author illegally uses huge verbatim passages of your unpublished work in a new novel. As part of your legal argument against the infringing author, you want to prove that you possessed your unpublished work decades before the author claimed to have started work on the infringing novel.
The truly serious issue with many types of poor man's copyright is that it simply doesn't prove what it intends to prove. If you have a sealed envelope that is postmarked with a date, that only proves that the envelope passed through the postal system on that day. It says nothing about what the contents of the envelope were at the time; notably, the envelope could be mailed unsealed and then filled and sealed at a later date.
More involved forms of poor man's copyright, like storing a work in an undisturbed bank safe deposit box, might carry more evidentiary weight, but such an approach could still be difficult to verify reliably: we must ask the bank to verify the negative that no one ever accessed the deposit box. This may or may not be something the bank is prepared to do, and they may or may not maintain such records perpetually.
If you really want to prove that you had possession of a creative work on a particular day (which, again, is not a complete legal argument, but may be helpful), you will need to find a trustworthy third party who can demonstrate to the court's satisfaction that the work has remained undisturbed in their possession since a particular date. A bank might be good at this, but the best agency I know of to carry out such a purpose is the U.S. Copyright Office! They allow you to register your work (for probably much cheaper than a decades-long deposit box rental) and confer the benefits of registration, e.g. statutory damages.
I don't know of any legal cases that have tested a poor man's copyright in court. On a personal note, I suspect there are none, because the primary case that poor man's copyright is intended to protect against -- someone taking your unpublished work and making the baldfaced lie that it is their own, such that your only recourse is to prove a timeline that renders their claims impossible -- seems somewhat uncommon. Consider the likelihood of such a case occurring combined with the likelihood of a plaintiff who has undertaken poor man's copyright, and I suspect the likelihood is small indeed.