No, you can't gain additional classes.
While you can take the feat Multiclass Mastery, it doesn't give you the ability to multiclass into more than one class if you don't already have that ability (for example, from being a Bard as your main class and thus having Multiclass Versatility). Multiclass Mastery does not say it allows you to ignore the prerequisites for the feats you take, which includes not having a class-specific multiclass feat for a different class, so you don't qualify for any that aren't for the Bard class.
However, you can take the feat and get some benefit from it.
Although you can't gain additional classes, there are still a few options left to you. After you've taken a class-specific multiclass feat, you no longer qualify for most multiclass feats, but you can still take other Bard multiclass feats. So whichever feats of Bardic Ritualist, Bardic Dilettante, and Master of Stories that you haven't already taken are options, as long as you meet their prerequisites.
Further, the power-swap feats Acolyte Power, Adept Power, and Novice Power count as multiclass feats (and are obviously available when you have a class-specific multiclass feat), so they are also options.