Breakdown of Er kann sich an dieser Scherbe verletzt haben.
er
he
haben
to have
dieser
this / that (fem.: diese, neut.: dieses)
können
can / to be able to
Questions & Answers about Er kann sich an dieser Scherbe verletzt haben.
Why is it kann ... verletzt haben and not hat ... verletzen können?
This sentence makes a present guess about a past event (It is possible that he injured himself), which requires a subjective modal. You form this with the present tense of the modal (kann) plus the Perfekt Infinitive (verletzt haben). The alternative hat ... verletzen können is an objective modal in the past, meaning he had the physical ability or opportunity to injure himself, which is not what we mean here!
Could I use könnte instead of kann here?
Yes, Er könnte sich ... verletzt haben is also completely correct. Using the Konjunktiv II form könnte just makes your guess a bit more doubtful or hypothetical (He might have injured himself), whereas kann indicates a slightly more realistic possibility. Both are commonly used for past assumptions.
Why do we use the dative case in an dieser Scherbe?
The verb sich verletzen (to injure oneself) pairs with the two-way preposition an to show what caused the injury. Because the shard is a fixed point of contact where the injury occurred, rather than a direction of movement toward a destination, an triggers the dative case here (dieser Scherbe).
Why does haben go at the very end of the sentence instead of right after kann?