So, if I'm at home on WiFi, in the morning typically there's nothing saying new messages for that Account. I manually do a refresh, and I have items (usually mostly just spam, but still messages fetched).
Account Options:
Scheduled Sync is checked
Include in "Sync All" is checked
Push main is checked
Use WiFi only is unchecked
Messages to sync is 21 days
Folders:
Inbox - Sync (plain folder), Push mail
Special Settings:
all off, so nothing special for this account
Global settings:
Mail, receiving:
System settings - Sync even with "Auto-Sync" is off
Messages to sync - 250
Scheduled Sync - checked
Use WiFi only - unchecked
Mail check interval - 10 minutes
Custom check interval - Not specified
Time reference point - 00:00
"Settle in" delay - None
When the app is active - unchecked
Only when screen is off - unchecked
WiFi in sleep mode - unchecked
Push Mail - checked
Check mail on events - Event Types - none