This paper arose from an attempt to bound the number of 2-planes contained in a smooth projective 4-dimensional hypersurface X. The idea is to follow a line of reasoning, originally attributed to Salmon, whereby one proves that the space of lines "highly tangent" to X has expected dimension. We prove this type of result for the space of lines which meet X to order 5 at some point. As a result, we obtain an upper bound of O(d^4), where d is the degree of X. We suspect that the space of order 6 contact lines also has the expected dimension, but we are not yet able to squeeze that out with our techniques. This would give an O(d^3) upper bound, which is best possible because the Fermat hypersurface contains 15d^3 2-planes.