Sure. It depends on your type of comp and the Ratio and (maybe even more importantly) Attack, whether you control these parameters or whether the settings are "baked in" by design.
If you use the compressor first, you may find that you like a higher gain setting on the Barbershop. I say this because the Barbershop can start to get farty and quacky at higher gain settings or when it peaks out from input gain or aggressive playing. Using the comp first does seem to smooth out the fartiness, and if what you're looking for out of the Barbershop is a more consistently overdriven medium-gain tone, you might like it better after the comp. In my case, I really like setting the gain low, with a little sag, and then really hitting the strings hard to get the tone to dirty up. I have come to think of that slight gruntiness on the attack as part of the charm of the Barbershop and part of what gives it that character that reminds me of the isolated bass tracks from classic rock and R&B recordings.
You might ask, why use a compressor after the Barbershop at all? To my ear, and at least at low gain, the Barbershop doesn't compress a lot on its own compared to other Overdrives (for example, those of the Tube Screamer lineage). I haven't measured this with an electronic tool, but I found I like a little compression after the Barbershop just to smooth things out and keep the playing nuances audible in the mix, as I would with the uneffected tone.
Again, all of this depends a lot on the kind of compressor you are using, and the settings, etc. Judging from the comments on various threads, many users seem to prefer the comp before the Barbershop, and I don't presume to know better. Also, I haven't tried a lot of comps with it, and the one I'm using at the moment is not one of the favorite TB reference comps (it's an older design Fishman Pro Platinum Bass, which is supposedly an optical design; I also tried a Bearfoot PGC, which I didn't like as much). So take my perspective with the appropriate grain of salt...