This three-stop route works because each place does a different job: Bangkok for the capital, Chiang Mai for northern culture and Phuket for an Andaman coast base. It fails when every transfer is treated as a sightseeing day.
Use at least two full days in each place after arrival, then add transfer days. Bangkok and Chiang Mai connect by air, train and bus; Chiang Mai and Phuket are usually most practical by direct flight. A connection through Bangkok can consume most of a day.
Open-jaw international flights can reduce backtracking, but only when the fare and baggage rules still make sense.
Northern smoke and PM2.5 can weaken the Chiang Mai part of the route. Phuket’s southwest monsoon, broadly May to October, can disrupt sea trips even when the island remains visitable. Check both conditions before locking non-refundable stays.
In rough-sea periods, make Phuket Old Town, food and land-based days part of the plan rather than emergency filler.