The I-V characteristics of CoFeB/MgO/CoFeB magnetic tunnel junctions show pronounced nonlinearities which are relevant both for sensor applications and for the basic understanding of spin-dependent tunneling. To study the relation between the tunnel characteristics and the tunnel magnetoresistance (TMR) ratio, a series of CoFeB/MgO/CoFeB junctions was annealed with stepwise increasing annealing time at different temperatures. The related TMR ratio and the I-V characteristics were measured in the temperature range between 15 K and 300 K. This allowed the comparison of I-V characteristics of the same junction for TMR ratios between 25% and 150% at 300 K thus eliminating the influence of variations in the preparation process of separate individual samples. In addition to a zero bias anomaly observed in particular at low temperatures and for large TMR ratios, a conductance anomaly in the I-V curves was observed around a bias voltage of 350 mV. A general correlation between the deviation from Ohmic I-V characteristics and the TMR ratio was found both for parallel and antiparallel magnetizations of both ferromagnetic layers. This means that the shape of the I-V curves directly scales with the spin polarization of the tunneling current and the proportion of coherent electron tunneling. Both the 350 mV conductance anomaly and the correlation between non-Ohmic characteristics and the TMR ratio can be explained by considering the contributions of the relevant majority and minority spin bands of the ferromagnetic contacts.