If I have a proof that someone has (or has had) an illegal copy of my work how much time do I have in order to sue for copyright infringement?
Copyright infringement has a 3-year statute of limitations. No mention of when the owner became aware of infringement.
§1323 · Recovery for infringement
(c) Statute of Limitations.—No recovery under subsection (a) or (b) shall be had for any infringement committed more than 3 years before the date on which the complaint is filed.
Typically civil suits (in the U.S.) would be actionable up to a year from the date the plaintiff (the person making the allegation) became aware of the issue, regardless of the actual date the action actually occurred. If in 2015 your boss passes you up for a promotion based on a legally protected status (race, religion, sex, ect) but the first time you hear about it is from a gossipy co-worker who repeats your bosses discussion of this reasoning, then you have until the same calendar day in 2019 to file a suit with the courts.
Are you suggesting the plaintiff could postpone the date they testify to "first becoming aware" after the date they were in fact aware (but have no proof of awareness), it is rather fuzzy. Say in our scenario, the victim knew when they were denied that the boss was somewhat hostile to them, but had no reasonable reason for the cause of that status (in this scenario, it's protected class) such that the victim can't tell if the boss is a bigot or just a jerk, until 2018 when the victim overhear's office gossip, it's technically valid postponement because they only realized the illegality of the situation. It's not illegal for the boss to be a jerk, as any bar patron in America will tell you during Happy Hour (whether you want to hear their story or not).
However, if it was clear to the victim in 2015 and they only brought the issue up in 2018-2019 window, then it may be... but there could be a defense that they source in 2015 was different and said this about every decision against someone who wasn't of the same protected classes as the boss... and the 2018 source heard a more reliable conversation (likely the boss flat out admitting it) and the victim only heard it then, making it more likely than not than the "boy who cried wolf" in 2015.
It's going to be a case specific determination of Justiciability (the ability of the courts to exercise legal authority) by the Judge and would not be an issue to bring up before a jury (when you hear that the Judge is "Trier of Law" and Jury is "trier of fact" it's for the judge to determine what the law says, here the question being "is the timeline within the statute of limitations?", not the jury) so if the trial is underway a lawyer would want to motion to dismiss the case. At any point during the trial, before the verdict is read, the defense may make a motion to dismiss the case, and the judge will issue his ruling. It rarely happens, but it never hurts to ask (in criminal law, SOP is to ask as soon as the prosecution rests it's case and prior to the defense's case. If granted at this point, it's because the crimes alleged were not given any evidence to ever happening).
After the jury verdict this is the subject of the appeals court. In civil law, the losing party may initiate an appeal, regardless of side, where as only the defense may initiate an appeal in a criminal case (Double Jeopardy is a fancy way of saying Prosecuters may not make appeals if they lose... and the winning side would never make appeals because if it was in error, it's still an error for your side.). Appeals are always on matters of law, never fact, so they do not rely on any new evidence. If there is new evidence, the appeal is that the evidence was not available at trial and had it been, it would have changed the questions of law the trial court would have been answering. The appellent court would then vacate the ruling, which means both parties pretend the trial never happened and we go back to the begining of the facts. Since the suit was filed on a specific date but "no trial happened wink wink nudge nudge it doesn't violate the limitation of a Year to Date.
Finally, because Lawyers get jokes about everything, you will almost never see a date solidly ascribed (On 1/1/18 my client was made aware of...) but rather nebulously ascribed (On or around 1/1/18 my client was made aware of...) which makes the timeline fuzzier because a general ballpark date is given. It could be 12/31/17 at 59.8 seconds before midnight... it could be January 2nd. It could be Christmas Day. It keeps the defense from pulling a technical lie by citing that it wasn't that date because they had crossed the date line and it was really tomorrow. It also helps for the witnesses to recall a general period of time for long past events. I might not remember that the weekend prior I saw the "Cherry Blossoms in D.C." then one can reasonably say that the events took place "In or around April 2018" because the Cherry Blossoms have a narrow window when they are in bloom, usually the first week or two of April, so it keeps the affidative from creating a perjury. If you read enough legal documents, you won't take long to see that every lawyer writes like this.