  We experimentally study tunneling of Bose-condensed $^{87}$Rb atoms prepared
in a quasi-bound state and observe a non-exponential decay caused by
interatomic interactions. A combination of a magnetic quadrupole trap and a
thin $1.3\mathrm{\mu m}$ barrier created using a blue-detuned sheet of light is
used to tailor traps with controllable depth and tunneling rate. The escape
dynamics strongly depend on the mean-field energy, which gives rise to three
distinct regimes--- classical spilling over the barrier, quantum tunneling, and
decay dominated by background losses. We show that the tunneling rate depends
exponentially on the chemical potential. Our results show good agreement with
numerical solutions of the 3D Gross-Pitaevskii equation.
