Most exchanges have rules that allow trades to be price adjusted or busted under certain but narrow circumstances.
Often these are categorized as obvious errors or clearly erroneous, or catastrophic errors. The exchange rulebook sets out the procedure and criteria.
For example on NASDAQ (Equities) the process is set out in their Rulebook under Rule 11890.
These occur when the person on the other side of the trade claims that the trade was clearly erroneous or an obvious error, and they submit a request for review to the exchange within the time window - usually within 30 minutes. The exchange then reviews the trade and makes a determination on whether the trade is to be busted or price adjusted. It is very rare for the process to take more than a day. The price criteria are quite extreme, so it has to be a big price jump.
Usually trades against customers will be busted while trades against non-customers will be price adjusted.
When a bust is determined by the exchange, both parties have their trades canceled and the money is returned as though the trade never happened. The bust should be reflected in the time and sales data.
Typically market participants are notified that the trades are under review and later on in the day the participants to the trade are advised if their trade (or their customer's trade) will have action taken.