Could I use both a colon and semicolon in a sentence?

Punctuation is very much a convention of signals which guide the reader as to how the sentence is to be read and construed. One might consider a comma as a one-beat pause, a semi-colon as two, a colon as three, and a full-stop four; but they also hint at the intonation in the same way as a question mark does.

Whilst punctuation is just a convention, there is nothing per se to stop one from using both a colon and a semi-colon in the same sentence, if that is what is truly necessary in order to guide the reader along.

In your example sentence, however, I would question the use of a semi-colon to join the two main clauses together. To my mind, one should be able to substitute "and" , but this does not work for me. For example: "Roses are red; violets are blue." Here, one can substitute "and"; the sentences have parallel structures, which helps.

Did you mean something like:

Colleges provide help for students who are struggling. They offer guidance with study skills to keep on top of coursework and homework, advice from counselors on dealing with the workload, and the option of dropping a class early.

Just one small point: you should keep the same grammatical form for the listed items after the colon. You have noun, clause, noun; so better would be: study skills etc, counsellors who will give advice etc, and the option etc.

I don't think there is any reason why you can't use both in one sentence, but in this particular case I would probably go for two sentences as your second clause is quite long.

The only rule I would instinctively follow is that one should not have two colons in the same sentence. Is that generally felt to be the case?

My understanding is that in the Middle Ages, European people generally read everything aloud; punctuation was introduced to help them do that and we have kept it because we read "aloud" in our head. Not every language is like this - Thai leaves no spaces between the words; older Arabic has no punctuation.

When writing in modern English, the whole layout, including whitespace between words, whitespace between paragraphs, whitespace indentations, and also punctuation in general - this all is used to help the reader. Recent research has shown that a European reader focuses on just a few letters at a time - something like the last letter of the previous word and then the next eight or so letters - apparently this is all the retina can take in at one time. The eye then jumps to the next group of letters. It is this scanning process that limits reading speed, not the conversion from letter symbols to meaning.

Conventionally, a colon marks a longish pause with a flat, wavy, or slightly rising intonation; but definitely without the falling intonation associated with the end of a sentence. If that is how you would say it, then a colon may be the right way to go - there is no particular rule as to how often.

For example: "Roses are red: violets are bluish, and come in three colors: dark blue, violet and indigo."
Note that here we have used a colon in two different situations: between two independent clauses; and to introduce a list.