Ok, as an experiment, could you remove an ID that you know is showing. The purpose would be to see if the WordPress Shortcode API has a text limit on shortcode options.
Another item I just noticed. The id option is using fancy quotes instead of straight quotes. WordPress doesn’t support the use of fancy quotes in the shortcode. Try this one, I removed the fancy quotes:
[connections id='1,5,6,7,8,11,12,14,21,27,44,45,46,69,70,71,73,75,79,82,83,84,85,86,87,88,89,90,91,92,93,94,95' page_limit=50 width='600' order_by='city' enable_category_select=FALSE enable_search=FALSE enable_map=FALSE str_work_phone='Tel:' show_addresses=FALSE str_bio_show='' str_note_show='' image_width=160 image=siteshot]