Why can't carbon form an ionic bond?
$\ce{C^{4+}}$ ions:

Single ions are really only observed in the gas phase.  There is absolutely nothing that prevents a C4+ ion from being generated in the gas phase, given sufficient energy.  Each successive electron removal requires additional energy though, so by the time you get to 4 electrons removed, you're looking at quite a lot of energy required.  Note that this does not mean that $\ce{C^{4+}}$ ions are seen in compounds, just that they can be made in the gas phase.  Any cation you can imagine can be produced in gas phase.

The difficulty arises when we want to do something with that carbon cation.  If I want to make a compound with it, it can bond with other substances and make new bonds.  Once that is done we can evaluate those compounds to tell whether the bond is truly ionic or not.  In condensed phases or in solution, we look at how the parts of a compound interact with the particles surrounding them to decide whether to treat them as ions or not.

The best chance at making a C4+ ion would be to bond it to fluorine, which is an extremely electron-hungry element.  When I actually try this, I find that the C-F bond is very polar, but the $\ce{CF_4}$ molecule that forms behaves as a covalent compound normally would.  For example, it does not dissociate into ions when dissolved or melted and it doesn't form an ionic crystal in its solid form.  As much as fluorine wants electrons, it doesn't want them enough to completely steal 4 of them from carbon, because the energy required to remove each electron becomes greater than the energy required to remove the one before it.  It never gets to the point of $\ce{C^{4+}}$ and $\ce{F^{-}}$ ions being bonded.

Lead can form a +4 ion, $\ce{Pb^{4+}}$, in compounds such as lead fluoride.  The reason that this can happen but carbon cations cannot is that the outermost electrons in a lead atom are much farther from the nucleus than those of a carbon atom and are easier to remove as a result.  

$\ce{C^{4-}}$ ions:

In gas form this does not happen.  Carbon's first electron affinity is positive, so we would expect to see it form $\ce{C-}$ ions in the gas phase if given free electrons to attract.  Adding a second electron to this substance would require it to attract an electron against its already negative overall charge, and that is not energetically favorable.

In order to make the $\ce{C^{4-}}$ion in a compound, a species would have to donate 4 electrons to carbon.  The attraction of carbon's nucleus to other atoms' electrons is just too low for that to happen.  The best candidates for a cation in an ionic compound of this type would be either lithium or cesium, however, when we try to actually make this compound we get another class of ions, the carbides, which feature $\ce{C2^{2-}}$ ions.  In short, it doesn't happen. 

Phillip's comment regarding the carbides is a good one.  There are a few metal carbides that feature carbon atoms bonding to a metal in the ratios we would expect if it were purely ionic, like $\ce{Mg2C}$.  In these compounds the carbon does have a formal charge of -4, but the properties of the substance are such that it behaves like a covalent network compound rather than an ionic compound.

What is the limit of this process?  Anions with -3 charge, like nitride or phosphide, do exist, though they require a very weakly electronegative species, like an alkali metal or alkaline earth metal, to form an ionic compound.

Carbon atoms can carry a charge as part of an ion, but only when they are part of a larger species (again, like $\ce{C2^{2-}}$) and they do not carry 4 additional electrons per atom.  Check out cyanide as another good example of this type of polyatomic ion.