To answer, we need to go beyond Lewis Theory. From a Lewis point of view, there is no reason why carbon can't form a quadruple bond satisfying the Octet Rule and leaving no electrons for further bonding. But it implies that C2 is a perfectly stable molecule, like N2, and that just isn't the case.
If we go on to the valence-bond model, in which bonds result from the overlap of atomic orbitals, we see a better explanation: carbon cannot form a quadruple bond because it doesn't have enough atomic orbitals pointing in the right directions. sp hybridization leaves two p orbitals over, while sp2 hybridization leaves one p orbital.
Valence-bond theory predicts two possible bonding states for C2:
1. double bond with all electrons paired, :C=C: (as seen in Alkene)
2. a triple bond with two unpaired electrons, .C(Triple)C. (As seen in Alkyne)
These are called resonance structures, and both must be considered as partial representations of the real situation.
Note: There exist a C2 molecule with bond order 2 with two  pi bond