Well, does it look like the app is completing the loading of those messages from the server (at those times)?
I.e. there is a spinning progress "wheel", the blue "@" status bar icon, etc.?
If that's the case...
Aqua's default settings for how much of each message it loads when checking mail are very conservative. In addition, sometimes mixed (html/plain) messages "fit into" those limits for the plain text part (so you see some text in the message list), but the html part, with html being more verbose, doesn't get to actual message text.
Please see app settings -> network and increase the values under "load while checking mail" for WiFi / mobile as appropriate.