There are two bars for your b-spec Bob. Career and experience.
Career only builds during races done through the b-spec menu in GT Life. (It rises by around 0.2% each race). This is what needs to max up to get the 'Air of Experience' trophy. When you do max him out, his experience quickly drops.
Experience is your b-spec Bob's 'race craft'. It rises in both b-spec and remote races. The more races you do, the more experience he gets.
Dont get experience confused with xp. xp is what YOU get, from having your bob's take part in races.