I would guess April and May are the best times, my dad and I went out to the Caney Fork sometime in mid-May and absolutely had a ball. We didn't keep track but we caught fish every cast (spin fishing, I went out on a limb to learn fly fishing). I caught a monster rainbow that day, 22" and probably 3 pounds. The next day they didn't generate and I had to go to school so my dad took my brother with him (he was back from college on break) and my dad caught a 24" brown that was pretty fat.
*I was pretty tired, took 15-20 minutes to get him in*
And the brown my dad caught on Powerbait, browns usually don't take Powerbait