CX bikes can be good all 'rounders, or they can be purpose-built racing machines. The description is all-encompassing, which makes generalizations difficult.

Mine's an all 'rounder. It fits tires to 38mm with full fenders. Others only 32 or so.

I run 28mm road tires in the three seasons, switching to 34mm CX tires for off-road excursions, and 35/38mm studded snow tires for winter. I could run skinner in the three seasons, but it's my primary commuter and I prefer the 28s on it. I have three other bikes for 25mm.