Two approaches to bounding the partial auto- and crosscorrelations of binary sequences are considered. The first approach uses the discrete Fourier transform and bounds for character sums to obtain bounds on partial autocorrelations of m-sequences and on the partial auto- and crosscorrelations for the small Kasami sets and dual-BCH families of sequences. The second approach applies to binary sequences obtained by interleaving m-sequences. A bound on the peak partial correlation of such sequences is derived in terms of the peak partial autocorrelation of the underlying m-sequences. The bound is applied to GMW, No (1987), and other families of sequences for particular parameters. A comparison of the two approaches shows that the elementary method gives generally weaker results but is more widely applicable. On the other hand, both methods show that well-known sequence families can have favorable partial correlation characteristics, making them useful in certain spread-spectrum applications.