That is potentially a clue. But in itself would not answer to a legal certainty (that one fact) that a bill with a serial number in a nefarious transaction means the person who withdrew it (from the ATM) made the transaction.
1) I can withdraw $300 from an ATM and purchase a coffee (using $5 of that). That $5 might be refunded as change to someone who then uses it (to buy chinese food) and then as change to another person. And that person purchases something that they shouldn't.
2) Of course at scale with large amounts (say $300 withdrawn 10 times to equal $3000) used in a nefarious transaction yes they could trace back but they still wouldn't have proof that the person withdrawing it didn't pass it to someone else (who then used it). Further they can't require you (in the US) to disclose what you did with the money after you withdrew it. However they might try to convince you you have to answer and tell them.
3) That said of course like with anything again yes it would point them in some way to you and then might be able to find another way to then focus on you and get what they need (in order to get a warrant to do a search or get other records released).