U₃Ni₃Sn₄ and U_2.9Ni_3.0Sn_3.9 single crystals exhibit a non-Fermi-liquid susceptibility T^-0.3 between 1.7 and 10 K. The electronic heat capacity coefficient (T) of U_2.9Ni_3.0Sn_3.9 varies as the square root of temperature between 0.3 and 5 K. Although most available non-Fermi-liquid models are in disagreement with these results, the heat capacity data are consistent with a renormalization group calculation for magnetic fluctuations near an antiferromagnetic quantum critical point (QCP). Alternatively, both the magnetic and heat capacity data can be fitted to a Griffiths-phase model for magnetic clusters near a QCP, using a single characteristic exponent = 0.7.