This is an interesting question because the answer is an absolute, positive maybe. Let me restate the general rule first: check the statute of limitations for your case with an attorney as soon as you think you may have a case. There are some very short statutes and you do not want to miss it because you waited too long!
Now, can you extend it? Yes, it is possible, but not very likely. Sometimes, an insurance adjuster will tell you that you have a certain time within which to settle your case or file a lawsuit. However, they then may tell you that this can be extended and they would agree to it. Great, but not helpful. The insurance company cannot extend the statute of limitations. The tortfeasor (horrible legalese for the person who caused your damages) can extend the statute in writing. But, in all of my years doing this I have seen it a grand total of ZERO times.
My advice: check with an attorney as soon as you think you have been injured or suffered damages. Find out what the statute of limitations is and follow that. Do not listen to anyone else!