Can I put a smaller breaker on a bigger wire?

Yes, and it is common. Section 240.4 requires conductors to be protected against overcurrent in accordance with their ampacity. It sets a ceiling, not a floor. A 10 AWG copper conductor on a 20 ampere breaker is fully protected because 20 amperes is well below the conductor ampacity. There is no code section anywhere that requires the breaker to match the conductor size going the other direction.

Why you would do it on purpose

Voltage drop is the usual reason. The informational note to 210.19(A) recommends keeping branch circuit voltage drop to 3 percent, with a combined feeder and branch circuit total of 5 percent. On a long run you upsize the conductor to hold that number while the load and the breaker stay the same. A 20 ampere circuit 200 feet out to a shop might need 8 AWG for voltage drop and still land on a 20 ampere breaker.

Future capacity is another. Pulling 6 AWG to a subpanel location now and protecting it at 40 amperes today lets you go to 55 amperes later with a breaker swap instead of a new pull. Also common is standardizing on one wire size across a job to reduce reel changes and mistakes.

What actually limits you

Terminations. Section 110.14(A) requires the terminal to be identified for the conductor being landed, and 110.3(B) requires you to follow the listing. Every breaker lug carries a marking such as 14 through 10 AWG Cu, or 14 through 4 AWG Al/Cu. Put 6 AWG under a lug marked to 10 AWG and you have a violation regardless of the ampacity math. When the conductor is too large for the breaker lug, the fix is a listed reducing splice, a polaris style connector, or a set of pigtails in the panel gutter sized per 240.4(B) and landed under the lug.

Box fill and raceway fill go up. Chapter 9 Table 1 caps a raceway with more than two conductors at 40 percent fill, and 314.16 box fill counts by conductor size, so 8 AWG at 3.0 cubic inches eats a box faster than 12 AWG at 2.25.

Grounding conductors ride along. Section 250.122(B) requires the equipment grounding conductor to be increased in size proportionally to the circular mil area of the ungrounded conductors when they are upsized for voltage drop. Upsizing the phase conductors and leaving the ground at the 250.122 minimum is one of the most common failures on a rough inspection.

Field takeaway

Bigger wire on a smaller breaker is always allowed. Confirm the lug range on the breaker, confirm box and raceway fill, and upsize the equipment grounding conductor per 250.122(B) if the reason you upsized was voltage drop.