Yes, both these adjectives are simultaneously possible. That’s why they call it a tragicomedy. But what’s interesting about the tone is that isn’t just bleak and comic; it’s bleak because it is comic, and it is comic because it is bleak. The common factor here is absurdity. Life is comic because of the absurdity of talking about turnips and carrots:
Fancy that. (He raises what remains of the carrot by the stub of leaf, twirls it before his eyes.) Funny, the more you eat the worse it gets.
With me it's just the opposite. (1.278-9)
But it's also bleak because men waste away their days talking about… turnips and carrots.