The electromagnetic dissociation of $^{59}\mathrm{Co}$, $^{89}\mathrm{Y}$, and $^{197}\mathrm{Au}$ target nuclei was inferred from measurements of cross sections for the one-neutron removal reaction. Beams of 2.1 GeV/nucleon $^{1}\mathrm{H}$, $^{12}\mathrm{C}$, and $^{20}\mathrm{Ne}$, 1.8 GeV/nucleon $^{40}\mathrm{Ar}$, and 1.7 GeV/nucleon $^{56}\mathrm{Fe}$ projectiles were used. Beam intensities were monitored using the $^{12}\mathrm{C}$(RHI,X${)}^{11}$C reaction, where RHI represents relativistic heavy ion beams. The experimental cross sections in excess of the estimated nuclear contributions are generally well described by use of the Weizsaaumlcker-Williams method for calculating the electromagnetic dissociation contributions but increase more slowly as the projectile charge is increased. Cross sections for one-neutron removal reactions are large for the heavier projectiles, ranging up to 707 mb. For these cases a high percentage is from the electromagnetic dissociation process.