We report measurements of the scalar and tensor static polarizabilities of the In-115 7p(1/2) and 7p(3/2) excited states using two-step diode laser spectroscopy in an atomic beam. These scalar polarizabilities are one to two orders of magnitude larger than for lower-lying indium states due to the close proximity of the 7p and 6d states. For the scalar polarizabilities, we find values (in atomic units) of 1.811(4) x10(5) a(0)(3) and 2.876(6) x10(5) a(0)(3) for the 7p(1/2) and 7p(3/2) states, respectively. We determine the tensor polarizability component of the 7p(3/2) state to be -1.43(18) x10(4) a(0)(3). These measurements set high-precision benchmarks of the transition properties for highly excited states in trivalent atomic systems. We also present ab initio calculations of these quantities and other In polarizabilities using two high-precision relativistic methods to make a global comparison of the accuracies of the two approaches. The precision of the experiment is sufficient to differentiate between the two theoretical methods as well as to allow precise determination of the indium 7p-6d matrix elements. The results obtained in this paper are applicable to other heavier and more complicated systems, and provide much needed guidance for the development of even more precise theoretical approaches.