Speaking Email currently can miss emails that are in the past when using Exchange server. This happens because we sync back only one day by default. If you only want to listen to your latest email this is not a problem. But if you expect to see emails from a few days ago, you will not see them.
We have now added a feature under account settings allowing you to choose how many days to sync back. Our sync defaults to one day, which means it checks for new mail since yesterday (i.e. it will not download mail from the day before yesterday).
If you get lots of mail and are up to date on reading it, this is a good setting. But if you don’t have heaps of mail every day, or want to listen to your inbox back a few days, you will want to set it to longer.