The arrest, the wrong warrant, and the clock running out
Every case ends one of three ways, and only one of them actually catches the crook.
Strip away the geography and history dressing and every case in the series resolves the same way: an arrest with a matching warrant in hand, a blocked arrest because the warrant names someone else (or nobody at all), or the deadline running out first. Only the first of those catches anyone.
Right place, wrong paperwork
Cornering the actual crook is not, by itself, a win. Every manual is explicit that an arrest without a matching warrant simply fails, and each game phrases the consequence a little differently: the original 1985 manual warns a player will be “thrown off the case!”, Where in the U.S.A. says flatly that “you won't be able to arrest the thief once you catch up with him or her,” and Where in Europe puts it more personally — arrive without the right paperwork and “the criminal will escape and you may want to reconsider your career as a detective.” None of the games let a wrong warrant end in cuffing the wrong person; the crook always slips free instead, whatever the wording.
When the clock wins
The second failure state doesn't require a mistake at all — just running out of time before the chase concludes. America's Past's Evidence Database puts it plainly: reach a 00 reading on the Time Remaining Indicator and “the villain will go free,” full stop, no partial credit for how close a player had gotten. Junior Detective keeps the same rule in softer language: leave the wanted poster unfinished when the GizmoTapper's power runs out, and the culprit simply escapes, even for the series' youngest audience.
The one case where the ending matters more
The capstone case — the one that finally names Carmen Sandiego instead of one of her gang — ends the same three ways mechanically, but only unlocks once a detective has already climbed most of the game's rank ladder. Bringing her in with a correct warrant is the closest thing the series has to a final boss fight, and it's earned entirely through the same clue-sorting and paperwork discipline every earlier case demanded — there's no separate mechanic reserved for her capture.