I asked this question through TI's technical support and the reply was to post here for information.
The LP3971 Power Management Unit does not always start. SYS_EN is high but Buck 2 and Buck 3 are not on. The application note at the end of the data sheet has a cold start power sequence. In this sequence there is a delay between SYS_EN and PWR_EN. In the documentation for the evaluation board, SYS_EN and PWR_EN are tied together. How important is the timing shown in the application note? The note only has typical times for most signals, no Max or Min.
What could be causing Buck2 and Buck3 not to start when both SYS_EN and PWR_EN are high?