Andrew Granville, in his beautiful article *Binomial coefficients modulo prime powers* (1997), attributes to Jacobsthal the congruence
$$
\binom{np}{mp} \Big/ \binom{n}{m} \equiv 1 \mod{p^{3+\mathrm{ord}_p\{nm(n-m)\}}} \quad \quad \quad (p \geq 5).
$$
In Chapter 7 of Alain's *Course in $p$-Adic Analysis* (GTM 198), the same congruence is attributed to Kazandzidis. 

This is a (mild?) generalization of Wolstenholme's theorem, $\binom{np}{mp} \equiv \binom{n}{m} \mod{p^3}$.

As both references explain, the general congruence is best understood via the $p$-adic analog of Stirling's asymptotic formula for the $\Gamma$-function. The $p$-adic logarithm of the left-hand side admits a convergent power series expansion involving the $p$-adic Bernoulli numbers - just as the higher order terms in the usual Stirling formula involve the usual Bernoulli numbers (or zeta values $\zeta(1-n)$). This is covered by formulas (35) and (38) in Granville's article. When $p \geq 5$, all terms in the expansion are seen to have at least the stated valuation $\delta := 3+\mathrm{ord}_p\{nm(n-m)\}$, and the only term with valuation $\leq \delta$ has the form $B_{p-3} \times nm(n-m) \times u_p$, where $u_p \in \mathbb{Z}_p^{\times}$ is a $p$-adic unit. 

Therefore the congruence holds modulo $p^{4 + \mathrm{ord}_p(nm(n-m))}$ if and only if $p \mid B_{p-3}$. Those are the Wolstenholme primes referred to by Greg Kuperberg in his answer.


[I came to this old question via the "Related" column for the new post http://mathoverflow.net/questions/195339/a-congruence-involving-binomial-coefficients by Richard Stanley, which proposes a different generalization of Wolstenholme's $\binom{2p}{p} \equiv 2 \mod{p^3}$. ]