The thermal rotation of iron nanoparticles dispersed in cyclohexane was studied by measuring the dynamic magnetic susceptibility above and below the freezing point of the solvent. Above the freezing point, the orientation of the magnetic dipoles changes mainly by reorientation of the entire particle. Below the freezing point, complete arrest of particle motion was expected, such that the magnetic dipoles would only be able to reorient themselves inside the nanoparticles (Neel relaxation). However, we find that thermal motion continues well below the temperature at which the bulk of the solvent is frozen. We ascribe this to local lowering of the freezing point, due to the presence of polymers in the close vicinity of the colloids. Furthermore, because strong dipole-dipole interactions result in the formation of dipolar chains, we have systematically studied the effect of particle size on dynamics in a frozen solvent. For the larger particles, our data indicate that local wiggling of the individual particles in a chain may become the dominating mode of thermal motion.