Are there any well-known examples of SHA-256 collisions? The popularity of SHA-256 as a hashing algorithm, along with the fact that it has 2256 buckets to choose from leads me to believe that collisions do exist but are quite rare.

Are there any well-documented SHA-256 collisions? Or any well-known collisions at all? I am curious to know.

I find that showing collisions to people I'm explaining hashing to is a great way to show them what non-invertibility means when they have a hard time seeing how the modulo_x operation relates to the SHA-256 operation.
No, there is not any known SHA-256 collision. Publication of one, or of a remotely feasible method to obtain one, would be considered major.

It is next to impossible that two distinct strings with the same SHA-256 have been computed so far. The most visible such computation is in bitcoin mining. By summing this data giving history of SHA256d hash rate (that's two SHA-256), I get that by end of April 2018 that had made 289.7 SHA-256, with the exponent growing roughly by 2 per year for the last few years. My computations, and opinion that they represent the bulk of SHA-256 made, have not been challenged there. Extending to 291 to account for other cryptocurrencies and (perhaps coverts) password search activity, odds are about 1 against 2256+1-91-91 = 275 that a collision occurred (see Birthday problem for cryptographic hashing).

For the purpose of illustrating collisions, perhaps make an example with SHA-256 restricted to its first 64 bits (16 hexadecimal characters instead of 64), and explain that each 2 bits added doubles the expected number of hashes before a collision occurs.