In addition to JonMark Perry's excellent answer it's also worth noting that many of us aren't wired the same as everyone one else.
When your dentist tries to numb a particular tooth, he or she is trying to numb the particular nerve, or group of nerves, that innervates that tooth. Those nerves may not be in the same location as the next patient. This blog by Nicholas Calcaterra DDS mentions this as the fourth of five reasons you may not be numbed. The most pertinent passage:
4. You’re wired differently
The human body is incredibly variable. People are double jointed. Remember the kid in grade school who could move his ears? Why is Usain Bolt faster then any other human? You get the picture.
Each person is different! Most people have what I might call “standard anatomy.” This means that the nerves going to your teeth are where you might expect them to be located. But just like Usain Bolt and the kid from fifth grade who could move his ears, some patients have extreme variability with the nerves going to their teeth. We see this most frequently with lower molars.
Some people may have up to 4 nerves going to their lower molar teeth. This can mean 4 different injections to get them numb! This doesn’t mean your dentist is incompetent – it means you’re wired differently. So if that happens to you, just think about Usain Bolt and the kid from fifth grade who could move his ears.
I had the same thing happen to me a very long time ago and just grinned and bore it; it was temporary, after all, and it has never happened since. My dentist then said he just couldn't locate the nerve going to that particular tooth. And in the citation above, if the dentist doesn't get all four of those nerves going to the molar in question, you won't be completely numbed.