Barding Inspiration has this rule (PHB page 33, emphasis mine):
…you use a bonus action on your turn to choose one creature other than yourself within 60 feet of you who can hear you. That creature gains one Bardic Inspiration die, a d6. […] the creature can roll the die and add the number rolled to one ability check, attack roll, or saving throw it makes.
If I roll an 18 and get a 2 or more on my Bardic Inspiration die, do I get a critical hit?
Note: This question was inspired by a ruling made by Jeremy Crawford that Cutting words can cancel a critical hit. He then later changed his mind, as explained in a different question. The answer to this question is still accurate, however (though it might require rewording), so I decided to keep everything here for posterity.
Best Answer
No, Bardic Inspiration can't cause a critical hit
Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.
Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.
The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.
The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.