To function, cells must move material internally. This intracellular transport is achieved by molecular motors, which transport vesicle-bound cargo along protein filaments. In vitro experiments have uncovered the mechanochemistry of how single, isolated motors turn chemical energy into mechanical work as they "walk" along a protein filament. In cells, however, multiple motors transport cargo. Some of these motors bind to the protein filament and contribute to cargo transport; others diffuse over the surface of the cargo, and the motors transition between the two roles. It is a challenge to experimentally determine the number of motors on a given cargo, let alone the time-varying number of these motors bound to the protein filament. To address this, we developed a method to estimate the number of motors and the number of filament-bound motors from position-time measurements of the cargo. The method uses our knowledge of the mechanics of the system and the mechanochemistry of the motors to infer the most likely number of all motors on the cargo and of filament-bound motors.