Internally, the bonding level is on a scale of 0 - 1000 points. The level shown is 0% - 100% in steps of 10%. You need to raise the bonding level over a 100-point threshold in order to see the percentage increase (by 10%.) Training gives you bonding points, but it's less than 100 points, so only sometimes you'll see the percentage increase.
example
start: points total 0, level shown 0%
use a bonding-level-increase item: points total 3, level shown 0%
perform some training: points total 53, level shown 0%
perform some training: points total 103, level shown 10%
play a game: points total 123, level shown 10%
play a game: points total 143, level shown 10%
use a bonding-level-increase item: points total 146, level shown 10%
use a bonding-level-increase item: points total 149, level shown 10%
use a bonding-level-increase item: points total 152, level shown 10%
perform some training: points total 202, level shown 20%
perform some training: points total 252, level shown 20%
(Note the amount of bonding points I have shown for various actions are just numbers I made up, to illustrate the example.)