Statutes of limitations (hereinafter "SOL") vary from jurisdiction-to-jurisdiction. If it is only one-year in New York (I've not confirmed this) that would not be surprising. SOLs exist for all civil matters and nearly all criminal matters.
I'd just like to point out that your question is not really limited to defamation or to the time frame for which the specific SOL runs for libel/slander ... at least as I've read it. It seems you are asking the broader question pertaining to what the philosophical or practical purpose(s) for SOLs in general are, as well as what effect these limits can have on the ability of a would-be claimant/plaintiff to get relief through the courts.
So, to answer whether a person who believes they've been defamed can still bring a cause of action after the SOL has passed: the short answer is no.
But, that is only the short answer. SOL is an affirmative defense, meaning that it can only be asserted if it is pled in the answer to a complaint. So, if John Doe feels he's been defamed by a libelous statement in the Times, and he files suit 4 years after the statement was made (and the SOL is 1 year), the court will still accept the filing of the complaint and Times must still be served. You do not lose the right to file your claim, when the SOL has passed, which is a very common misconception of the law. And, Times must still file an Answer to avoid defaulting. However, in that answer Times (no doubt by and through qualified counsel) will most likely assert a variety of affirmative defenses pro forma, which will include the passage of SOL (the claim is stale). If, in fact, the SOL has passed and the affirmative defense is pled, the next thing the Times will do is file a motion to dismiss. The judge will view the complaint in a light most favorable to Doe, the plaintiff, which will lay out all the facts (including when the libelous statement was published). If the cause of action accrued (this just means when the injury was sustained) and the claim was not filed prior to the expiration of the SOL, Doe's case will be dismissed. If the Times does not assert SOL though, it is deemed to have been waived for that and any subsequent related proceedings.
There are ways to toll (extend) the statute of limitations. There is a discovery rule (this does not mean Doe didn't read it until after the SOL, even though he was able to). It means that the SOL can be tolled in cases where one could not have discovered the tort occurred. In these instances, the SOL doesn't begin to run until the discovery was either made or should have been made.
This is most typically seen in medical malpractice cases (e.g., a surgeon leaves a sponge or instrument in your abdomen, and after years of failed treatments for IBS your doc sends you for an x-ray and an MRI and it's discovered) or in asbestos cases (you don't know you breathed it until you have asbestosis) – those types of scenarios. In cases like that, the SOL begins to run when you find out the tort occurred. This has also been successful in recovered memory cases where childhood sexual abuse occurred but was suppressed.
Incapacitation is another way the SOL can be tolled. So, in Doe's libel case, the SOL can be tolled if he is incapacitated (in a coma and didn't wake for 4 years, in a mental hospital in a break from reality for 5 years) – in such circumstances you can still file a claim and have it survive a motion to dismiss based on SOL once you are rehabilitated.
Also, Absence due to military service is reason to toll in some jurisdictions.
Lastly, (at least the last one I can recall) is that minors can often toll the SOL however long the statute is (say 1 year) after they turn 18.
As for why there are SOLs: That is more public policy than anything else. It is one of the oldest rules of law known to modern man, dating back all the way to early Greek and Roman law.
Statutes of limitations are a fundamental part of EU and US law, as well as most other modern legal systems. They function to prevent fraudulent and stale claims from arising after all evidence has been lost or after the facts have become dubious and unclear due to the progression of time, which can lead to lost or uncertain memories, death of witnesses (for each side), or disappearance/inability to locate witnesses.
The policies behind SOLs are also for judicial economy and to protect defendants from having a never-ending potential suit looming over them. Often, the seriousness of the crime or the tort, or some element of it, dictates the time frame of the SOL. It is a means to ensure that plaintiffs pursue their claims with reasonable diligence: I.e., if it matters to you, file it. Also, as I touched on above, time disadvantages defendants. Plaintiffs are the "injured" party, so they tend to retain evidence much longer than a defendant, who might not even realize he has done wrong – especially in civil matters.
So, in a libel case, the whole basis of the claim is the irreparable damage to your reputation. If more than a year has passed and you didn't know about it, one could assume the damage never rose to the level of having sufficed to make a valid claim in the first place. That is why they run fast in those cases.
They run fast in medical malpractice claims too, but this is a result of tort reform – the public policy that litigation against doctors/hospitals causes increased health care costs for everyone. So, policy suggests it's best for society to dispose of these claims quickly (not so much if you're the plaintiff). Whereas most regular negligence claims can have SOLs as long as six years in some jurisdictions where I have practiced. So there are clearly policy determinations going into these legislative acts.
The Golden Rule of Law that I tell all of my clients is this: If you think you have a claim, talk to a lawyer right away, because if you don't you can lose the right to pursue relief, much, much quicker than one may ever imagine!
I know this is a long answer, but I get asked this (type) of question all the time. Plaintiffs feel the SOLs run too fast, while Defendants cannot believe they can still be on the hook!