Is it possible to level up several levels at once in Ingress?

Your AP is a continuous value. In other words, when you reach 10,000 AP and go to level 2, your AP bar doesn't reset to 0. It stays at 10,000 and in order to reach level 3, you need 30,000 AP. This means that you don't actually need to earn 30,000 AP. Instead, you need to earn 20,000.

So yes, you would level up twice.

Yes, it is also possible (and quite common) another way, jumping from 12 to 14.

If you already have 17M AP, and two platinum medals (agent can have recharger and gardian quite early) but only 6 gold medals, you are lvl 12. When you gain another gold medal you jump to lvl 14.

In this case, oddly, your scanner first displays the lvl 14 level up, then the lvl 13 one.