Injuries happen in the playoffs, just like in the regular season. The difference is that teams are more cagey about the health status of their players than they usually are.
Both the Leafs and the Capitals have AHL teams about to start the first round of the playoffs, so neither have called up many players, nothing like the 12 extra bodies the Columbus Blue Jackets added recently. The Leafs called up Kasperi Kapanen on a regular, non-emergency recall, and they have two of those left to use if they want to. The Capitals actually sent players back to Hershey as the NHL playoffs started.