We study the motion of a probe driven by microtubule-associated motors within a living eukaryotic cell. The measured mean square displacement, <x(t)2> of engulfed 2 and 3 microm diameter microspheres shows enhanced diffusion scaling as t(3/2) at short times, with a clear crossover to ordinary or subdiffusive scaling, i.e., t(gamma) with gamma less than or equal to 1, at long times. Using optical tweezers we tried to move the engulfed bead within the cell in order to relate the anomalous diffusion scaling to the density of the network in which the bead is embedded. Results show that the larger beads, 2 and 3 microm diameter, must actively push the cytoskeleton filaments out of the way in order to move, whereas smaller beads of 1 microm diameter can be "rattled" within a cage. The 1 microm beads also perform an enhanced diffusion but with a smaller and less consistent exponent 1.2<gamma<1.45. We interpret the half-integer power observed with large beads based on two diverse phenomena widely studied in purified cytoskeleton filaments: (1) the motion of the intracellular probe results from random forces generated by motor proteins rather than thermal collisions for classical Brownian particles, and (2) thermal bending modes of these semiflexible polymers lead to anomalous subdiffusion of particles embedded in purified gel networks or attached to single filaments, with <x(t)2> approximately t(3/4). In the case of small beads, there may also be a Brownian contribution to the motion that results in a smaller exponent.