They would not stack as in your example. However, I would allow them both to be used in their respective applications. So for an attack roll or saving throw, someone with Bardic Inspiration and Bless can choose to add either a d6 or a d4, or both. Of course, only Bardic Inspiration can be applied for an ability check; Bless does not affect that.
EDIT: So yeah, what Naatkinson said.