In addition, what is the account setting with the email client used on the computer for accessing the account for removing messages from the server when downloaded? If set to remove messages from the server when downloaded and the email client is actively checking the account for new messages and downloading new messages received at the server while the iPhone's Mail client is doing the same, the messages downloaded with the email client on the computer will be removed from the server when downloaded so the messages will not be available for the iPhone's Mail client to download when the iPhone's Mail client checks the incoming mail server for new messages afterwards.
Does the account have an authenticated SMTP server?
If not, there will be problems sending messages with the account depending on the the internet service provider being used for the internet connection at the time.