Thermodynamic behavior of a mixture of weakly interacting fermions and bosons is investigated in 4-\ensuremath{\epsilon} dimensions by the renormalization-group method with the purpose of studying scaling and crossover properties of the system in the tricritical region. Conventional tricritical scaling, first found to break down for a classical infinite-component model, is seen to do so more spectacularly in the case of the mixture. Whereas in the infinite-component model, conventional scaling holds in the ordered and disordered phases separately (i.e., with different tricritical exponents), it is impossible in either of the phases of the mixture. The breakdown of scaling in the mixture is associated with the dimensionless strength ${v}_{6}$ of the six-point interaction in the effective Hamiltonian that causes the parameters of the renormalized Hamiltonian to depend on two combinations of scaling fields rather than one. The strength ${v}_{6}$ is a quantum-mechanical parameter, being proportional in three dimensions to ${b}^{3}$/${\ensuremath{\lambda}}_{T}^{4}$${K}_{F}$, where ${\ensuremath{\lambda}}_{T}$, ${K}_{F}$, and b denote, respectively, the boson thermal wavelength, the Fermi momentum of the fermion component, and the scattering length associated with the fermion-boson interaction. The square root of this quantity agrees with the nonuniversality parameter, which was found to characterize tricritical amplitude ratios in three dimensions in an earlier work.