Among the reasons survivors don’t go to the police, one is the well-founded fear they won’t be believed; another is that they may be accused of lying or even be prosecuted themselves. Cases where the accused and the accuser know each other, or where there is little physical evidence, are especially difficult to prove.

Even evidence doesn’t guarantee a prosecution. In the 24 states where there is no DNA exemption to the statute of limitations, if a rape kit finally is tested and there’s a match, the victim is out of luck if it has been too long.

It shouldn’t take multiple accusers to bring about a full investigation, but that’s sometimes the reality. The allegations against Mr. Cosby span decades. Once one woman speaks up, others may find the courage to do so. According to New York City crime figures, the number of women willing to report rapes that occurred in previous years is rising rapidly — a phenomenon that has been called “the Cosby effect.” Those survivors shouldn’t miss their day in court because they were too traumatized or too scared of possible repercussions to speak out sooner.

While good data is lacking on how many suspects are charged a decade or more after the fact, reports of such cases are rare. Some assailants, though, have gotten off because they admitted to their crimes after the statute of limitations expired.

The differing statutes of limitations across state lines create a ZIP code lottery. Where a person is assaulted shouldn’t determine whether she gets justice.

An end to statutes of limitations would also allow prosecutors more time to build thorough cases against alleged assailants, instead of rushing up against a dwindling deadline to file slapdash charges, potentially compromising the rights of the accused in the process. It would also allow the accused to defend themselves in a court of law, with its high bar to conviction, rather than in the court of public opinion.

Perhaps some state legislators fear ending statutes of limitations for sexual assaults would create chaos and a flood of reports of decades-old crimes. They need not worry: The 16 states with no statute of limitations for rape or sexual assault are not overwhelmed with huge numbers of stale claims.

Increasing reporting of rapes and sexual assaults is a complicated challenge, and requires cultural changes as well as improvements in law and policing. For state legislatures to eradicate statutes of limitations for sex crimes is a small enhancement, not a big fix. But for the women and men who find the courage to speak after years of silence and fear, it could be everything.