Isn't by definition Neptune not a planet either?
I beg your pardon?
Well, the current definition involves 3 conditions the last of which supposedly invalidates Pluto from being one.
A planet is a celestial body that
(a) is in orbit around the Sun
(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape
(c) has cleared the neighbourhood around its orbit
Condition (c) is what doesn't make Pluto a planet, but by it's orbit crossing with Neptune (again, condition (c)) Neptune status also becomes unclear. Of course Earth, Mars and Jupiter become also fuzzy, but by having it's orbit crossing with asteroids that don't come under any condition except (a) their status is clearer.
Anyway, personally I think this is not a scientific problem but rather a language one. Planets, dwarf planets and asteroid are too broad terms to designate stellar objects, as they have too many similarities between them.