It DOESN't affect what tama character you get, at least on V2 upwards. People think that if you miss training, you won't get a top tier character, but I've had a Mametchi with 2 bars. That's for V2 and V3.
Seeing as growth is determined by family and skills on V4 and V4.5, nothing happens if you have full training.
On V1, it may do something, but I don't have a V1, so I can't help you with that.