Why is H2O a weaker acid than acetylacetone? Shouldn't a hydrogen connected to an oxygen be more acidic than a hydrogen connected to a carbon atom?
Your instructor is right, up to a certain point. There are factors that are more important (the atom to which it is bound) and factors that are less important (resonance, etc.), and it's good to be able to order these relative factors. The devil is in the details, though. If your resonance stabilisation is strong enough (and in this case it is) then it can overcome the more important factor.
Before starting, let me point out that CH's are less acidic than OH's, but this isn't actually to do with electronegativity. If it was about electronegativity, then OH's should be more acidic than SH's as O is more electronegative than S, but the opposite is true. Same applies to HF and HCl: F is more electronegative but HF is actually a weak acid. It has more to do with the X–H single bond strength than anything, as a weaker bond is easier to break (homolytically). The X–H heterolytic cleavage, relevant for an acid–base reaction, is sort of like a homolytic cleavage, plus an electron transfer from H to X. The electronegativity comes into play in the electron transfer step, but it's just not so important a consideration, compared to the bond strength which describes the homolytic cleavage.

I'm giving pKa's to the nearest 5(ish) in this discussion. The exact numbers don't matter too much, we're more interested in broader trends.
If you compare water and methane, their pKa's are ~15 and ~50 respectively, which is a difference of about 35 units. That's roughly how much the adjacent-atom factor counts for, when comparing C and O.
If you then compare acetone and methane, pKa's are ~20 and ~50 respectively, which is a difference of about 30 units. Not quite as big as the previous one, which was 35.
Acetylacetone, though, has two carbonyl groups, that's twice as many as in acetone. Now, resonance stabilisation doesn't scale linearly in that fashion and hence the additional stabilisation over acetone is only ~10 units. However, it's still enough to bring the pKa down to ~10, which is more acidic than water.