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We present an analysis of the B- and V-band rise-time distributions of nearby Type Ia supernovae (SNe Ia). Drawing mostly from the recently published Lick Observatory Supernova Search sample of SNe Ia, together with other published nearby SNe Ia with data starting at least 1 week before maximum light, we use a two-stretch template-fitting method to measure the rise and decline of BV light curves. Our analysis of 61 SNe with high-quality light curves indicates that the longer the time between explosion and maximum light (i.e. the rise time), the slower the decline of the light curve after maximum. However, SNe with slower post-maximum decline rates have a faster rise than would be expected from a single-parameter family of light curves, indicating that SN Ia light curves are not a single-parameter family of varying widths. Comparison of the B-band rise-time distribution for spectroscopically normal SNe Ia to those exhibiting high-velocity (HV) spectral features indicates that HV SNe Ia have shorter B-band rise times compared to their spectroscopically normal counterparts. After normalizing the B-band light curves to Δm₁₅(B) = 1.1 mag (i.e. correcting the post-maximum decline to have the same shape as our template), we find that spectroscopically normal SNe Ia have a rise time of 18.03±0.24 d, while HV SNe have a faster B-band rise time of 16.63±0.29 d. Despite differences in the B band, we find that HV and normal SNe Ia have similar rise times in the V band. Furthermore, the initial rise of a SN Ia B-band light curve follows a power law with index 2.20^+0.27_- 0.19_, consistent with a parabolic rise in flux predicted by an expanding fireball toy model. We compare our early-time B-band data to models for the predicted signature of companion interaction arising from the single-degenerate progenitor scenario. There is a substantial degree of degeneracy between the adopted power-law index of the SN light-curve template, the rise time and the amount of shock emission required to match the data.