Like all rodents, rats' incisors constantly grow throughout their lives. This is so they can gnaw and nibble to their heart's content without having to worry they'll wear their teeth completely away. However, it does mean it's possible for rats' teeth to become overgrown. Rats can usually maintain the length of their teeth through natural ratty behavior, but you'll need to provide them with certain items.
Bruxing or Chattering
You may have noticed your rat making a grinding sound with his teeth. This is a normal behavior for a rat and is called bruxing or chattering. It's likely that your rat will do this expressly to help keep his teeth at the correct length, but he may also chatter when he's happy, stressed or nervous. If he does this constantly, it may be a sign that you haven't provided him with ample things to gnaw on.
Gnawing on hard wood will help your rat keep his teeth worn down to a comfortable length. Numerous different wooden rat toys are available commercially for this exact purpose. It's best to buy these specialist products, rather than just picking up a branch from outside, because you can be sure the wood's safe for your rat and isn't harboring any harmful bacteria or other nasties.
Treats and Stones
Probably your rat's favorite way of keeping his teeth worn down is by gnawing on food. Hard treats, such as low-calorie dog biscuits are ideal for keeping Ratty's teeth trimmed down to size. Mineral stones are another option. Not only do these stones contain some essential nutrients for your pet's diet, they're also hard enough to grind down his teeth when he gnaws on them. If you notice your rat gnawing on the bars of his cage, this isn't a sign that he's trying to wear down his teeth; it generally means he needs a bigger cage.
Unless your rat has a medical issue that stops his teeth from grinding properly, it's extremely rare that they'll become overgrown. However, it's important to check his teeth regularly to ensure they're of a good length. When a rat's teeth are too long, he may be unable to eat and could potentially starve. If you think your rat's teeth are overgrown, take him to see a veterinarian, who will be able to trim them down to size and find out if there's an underlying issue.