If you want help with it, could you provide more details:
What version of Android are you using? Is it Marshmallow? That one has dooze mode that kicks in at some point and could interfere with apps running in the background.
What Aquamail version are you using?
Have you tried keeping one account sync settings at 15 or 30 minutes, while having the rest as you described?
If that works, uout can try increasing that sync time gradually (30, 60, 90 ... minutes) to see when it breaks)
Alternatively, you can start with the default settings, and the increase the sync interval for just one account in 30 minutes increment and see when that would break (if it will).
First, you might find some reasonable settings that would still work for your needs. Second, it might give an insight into what is the culprit in your situation.
And, just in case, what specific settings are you changing in order to get what you described?
Settings -> receiving -> scheduled sync -- is it check-marked?
Is your setting Receiving -> System settings (receive even when system auto-sync is off) enabled?
Are you using any battery "optimizers", such as battery doctor or greenify?