On peut utiliser les toilettes? Does that make sense?

If I were in class in France, and I asked that, it is a polite way to ask to go to the bathroom?

What you said gets the point across, but their are better ways to say it. Raise your hand, get the attention of the proffesor and then use the options below:

Excusez moi? Je voudrais aller a la salle de bains s'il vous plait.

or

Excusez-moi? Est-ce je peut utiliser la salle de bains?

*** Feel free to substitute "a la toilette" in place of "a la salle de bains" - but I think la sale de bains sounds more proper.

est ce que je peux aller aux toilettes would be better

i would say Je voudrais aller au w-c, s'il vous plait. cause if you say est-ce que je peux aller its like can i go the bathroom of course you can the difference between may I and can I.

"Est-ce que je peux aller aux toilettes" when you want permission to go, "puis-je aller au toilettes" is a better way (more elaborate).
If you want to use the toilet of a place like a restaurant or even the toilet in someone's home, you should say "puis-je utiliser vos toilettes" or "est-ce que je peux utiliser vos toilettes".
:)

In my French class, we were taught to say, "Est-ce que je peux aller aux toilettes, s'il-vous-plait?"

It is correct, but a bit abrupt.
The 3 answers above mine are correct, apart that in the last one, you should use "aller aux toilettes" and "utiliser les toilettes".
A "salle de bain" is a room with a bath tub or a shower.

Actually you're translating it from the english : can I use the bathroom? Well, it's a lot better to say : puis-je passer aux toilettes?

Add "s'il vous-pla卯t".
And if they say yes, "Merci!".