Can I use multiple semicolons in the same sentence as long as they follow the rules for using semicolons?
In general, for use in a list - yes. For use in linking sentences - no. See below as to why:
In general, I use semicolons for two purposes: to link together two sentences that are related without having the pause you get with a period, and to separate things in a list where commas are being used. In this context, it'd be ok to use multiple ones in the list but not ok to use them to link sentences (this being the general rule - let me flesh this one out below).
In the list example, I could say something like:
I went to the pet store today and saw many animals that want a home: a blue, white, and orange parrot; a cute kitten, tabby kitten; a very playful, fun-loving puppy; and a small school of blue, red, and transparent fish.
The semicolons help to separate the items in the list from the adjectives describing each animal.
For the linking of sentences, let's see what happens if I use multiple semicolons:
It was a dark and stormy night; a night of crashing skies and lashing rain; the skies opened and poured forth Neptunian waters while Vespian fire flashed across the inky heavens;...
The feeling I get as I reread what it is that I just wrote is that it's a muddle - there is no pause for reflection, no place for the breath to be. If that is what you want your reader to experience, then go for it! But be aware that your writing may lose impact because there aren't those places to pause and breathe - that the reader may just shoot across those passages and not experience them simply because the eye is looking for the safe haven of a period.