Yes, every accessible category is well powered.

Since presentables are dense, they form a strong generator. Every category with a strong generator is well powered.

Some considerations, not a full answer (yet).

• In Accessible categories and models for linear logic, at page 2, Barr claims that every accessible category is well powered. He even claims that is observed in the classical reference by Makkai-Parè. I did not manage to find it.

• A locally small category with finite intersections of subobjects and a (strong) generating set is well-powered, this appears in Johnstone [Sketches of an elephant, Remark A1.4.17]. Thus when an accessible category has finite intersections, it is well powered.

I still believe that every accessible category is well powered, and I am looking through the literature.

• Observe that the statement that appears in Rosicky-Adamek on page 2, namely every locally small category with a strong generator is well-powered is wrong, as proved by the following counterexample.