Atomic radii are not really well-defined. There is a great deal of uncertainty associated with them, mainly due to the somewhat philosophical question ‘where does an atom end?’ Orbitals, in theory, extend into infinity for all (mathematical) intents and purposes, yet the probability of locating an electron in Far, Far Away is zero by all sensible rounding methods. However, the decrease is exponential, so you need to define a cutoff value if you want to know where an orbital ends to draw the beautiful pictures you probably have seen before.
This only gets worse when considering entire atoms rather than orbitals. We typically consider atoms as spherical (hence radius). But for every element except hydrogen, helium, lithium and beryllium (possibly including boron, too) there is at least one additional non-spherical orbital occupied. Can we still truly define an atom as a sphere? And even if we can, consider the case of transition metals; should we be talking about some cutoff value for the outermost s orbital or about one for the lower shell’s d subshell, whose electrons are considered valence electrons?
Now in theory, you could ignore all those philosophical discussions and just take the solid-state structure of an element, measure the shortest distance between two nuclei, divide it by 2 and call that the atomic radius. Well, that is what is done. However, many elements have different structures that they adopt or can adopt at room temperature and standard pressure. For other, especially non-metals, the shortest interatomic distance is not an atom-atom contact but an interatom bond. In a select few, the interatomic bond can have different bond order values; sixtytrees noted carbon whose most stable phase is graphite (bond order 1.5) but which also features a diamond allotrope with a bond order of 1.
The covalence problem can be overcome with a set of correction factors that introduce further uncertainties and are added differently by different sources.
All these issues mean that there is not one ‘true’ way to determining atomic radii; and since the methods differ, so will the results. It is only natural for some elements to be larger than others in one scale but smaller in another.
The whole problem gets infinitely worse with ionic radii. Ionic compounds exist as crystals with anions and cations — but where to the cations end and the anions begin? How do we deal with the fact that no bond is truly ionic and no bond between two different elements truly covalent?
Even if you determine e.g. the ionic radii of $\ce{Na+}$ and $\ce{Cl-}$ by various methods, (e.g. from comparing sodium fluoride, sodium carbonate, magnesium chloride, ammonium chloride …) and then add the two values together to arrive at an interionic distance in the $\ce{NaCl}$ crystal, it is very likely that your result is off.
