Yes, in some traditional cel animation only the part that's moving is animated while the rest of the figure remains static. This is typical of Hanna-Barbera's "high volume" animation style, and is sometimes referred to as "limited animation." The "pinnacle" of this limited animation style could be found in a lot of Japanese animation from the 1960s-Speed Racer is a great example.
"Full animation" uses a separate cel for the motion being drawn, but the entire figure (let's say Bugs Bunny) is drawn (and inked and painted) for each cel. Of course if part of that figure will not be seen, like he's partly hidden by a tree, then only the visible part will be drawn in the first place, but all of that character will be drawn for every cel.
And before anyone asks, "cel" is short for "cellulose" from the original material used for the transparent overlays-originally cellulose nitrate, they are now made from cellulose acetate.