Before iOS9, I often had 200-300 messages (some with attachments, though none all that big) in my inbox; some were weeks old. No problem reading them.
All that convenience went away with iOS9.
If the message isn't still in the server inbox (inbox, not just "on the server"), and my device isn't connected, then "this message has not been downloaded", or only part of it still shows.
I do not have messages deleted from server automatically from any computer or device. I manage it all manually (deleting, filing, spam-check, etc.).
Except that...
The whole problem is somewhat inconsistent -- not all the messages "disappear" (I haven't figured out what attributes allow them to "stay" or "go"). Just to make it more obvious that they are really in the device "somewhere" -- the preview lines are still readable in the list view! And flagged messages stay flagged, so at least I know what I had marked to deal with "later" even if I can't see the details.
I'm gradually adapting to my iphone mail not being any use other than a brief (maybe one-time) preview. IOW it's no longer any good as a reference to look back later, or a place to re-read long messages (e.g. some newsletter/updates) another day. Less time on phone, more time at desk.
If Apple does fix this problem with an update sometime (not holding my breath) I doubt I would ever feel as confident in the mail app as a useful and convenient tool as I have these past several years.