Let {X-n, n >= 1} be a sequence of independent random vectors in R-d, d >= 2, with common continuous distribution function F. For fixed n >= 2, we say that a multiple maximum occurs among the sample points X-1,..., X-n if M-n = X-i for some i = 1,..., n, with M-n the componentwise sample maximum. In this paper, we investigate the asymptotic behaviour (n -> infinity) of the probability of observing a multiple maximum by imposing an asymptotic tail condition on F.