Swift monitoring of NGC 4151 with an ∼6 hr sampling over a total of 69 days in early 2016 is used to construct light curves covering five bands in the X-rays (0.3-50 keV) and six in the ultraviolet (UV)/optical (1900-5500 Å). The three hardest X-ray bands (>2.5 keV) are all strongly correlated with no measurable interband lag, while the two softer bands show lower variability and weaker correlations. The UV/optical bands are significantly correlated with the X-rays, lagging ∼3-4 days behind the hard X-rays. The variability within the UV/optical bands is also strongly correlated, with the UV appearing to lead the optical by ∼0.5-1 days. This combination of ≳3 day lags between the X-rays and UV and ≲1 day lags within the UV/optical appears to rule out the "lamp-post" reprocessing model in which a hot, X-ray emitting corona directly illuminates the accretion disk, which then reprocesses the energy in the UV/optical. Instead, these results appear consistent with the Gardner & Done picture in which two separate reprocessings occur: first, emission from the corona illuminates an extreme-UV-emitting toroidal component that shields the disk from the corona; this then heats the extreme-UV component, which illuminates the disk and drives its variability.