Your problem description suggests you have a conflict between the table size (eg too large to fit on a single page), the 'Allow row to break across pages' setting and paragraphs formatted as 'keep with next' (prevents adjoining rows splitting across page breaks). You could:
• try removing the 'keep with next' attribute from all end-of-cell markers
• reduce the size of the top/bottom margins; and/or
• reduce the height of the content in various/all rows (eg use a smaller font, reduce paragraph before/after spacing, reduce table row top/bottom margings, etc).

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Cheers,
Paul Edstein
[Fmr MS MVP - Word]