Let f be meromorphic in the plane and analytic at 0. Then its diagonal sequence {[ n/n]} ∞ n = 1 of Padé approximants need not converge pointwise. We ask whether by reducing the order of contact (or correspondence) of [ n/n] with f at 0, namely 2 n + 1, we can ensure locally uniform convergence. In particular, we show that there exist rational functions R n of type ( n, n), n ≥ 1, and a sequence of positive integers { l n } ∞ n = 1 with limit ∞, depending on f, such that R n has contact of order n + l n + 1 with f at 0, and which converge locally uniformly to f. Moreover, for any given sequence { l n } ∞ n = 1 , there exists an entire f for which order of contact higher than n + l n is incompatible with convergence.