Yes, the messages are downloaded, but...
1) The number of messages downloaded is limited to the number configured in the settings (I believe, it is 100 messages by default)
2) Messages are purges beyond certain number (I believe 250 by default) and age (2 weeks?) - those options are also configurable.
3) Long messages may not be fully loaded, especially when on mobile network. Rather only first [?]kB (1?) is loaded. Again, this is configurable.
4) Attachments may or may not be loaded. It is configurable.
In particular, IIRC, you can specify to preload attachments on Wi-Fi.
There might be some other cases, and Kostya will probably add to that list.
From what you described, I believe your are experiencing #1. If you need more messages loaded, increase that number in the Settings -> [General] Messages to sync -> [it goes up to 5000]
Hope this helps.