Several things can affect Mail:
1. Old Mail plugins messing with a new OS X update.
2. Email server change that you are no aware of or the email server techs decided to do without consulting users.
3. You let your INBOX size to get WAY TO BIG and the server account is almost maxed off for your account.
Now in OS X of email there are two (three if you included Microsoft domain Exchange),
P.O.P &
IMAP.So in OS X Mail there are folders you save email in the section 'On My Mac'. This is were you can make folders that are local to your Mac to drag email from the Inbox to your Mac (DetaMac posted how to get to the folders without Mail).
My post earlier about 'connection doctor' is an old part of Mail so you as a user can see the steps Mail is doing to talk to your email server. It will help you, as a user, track down email server connection problems.
Now you heard about "optimizing" Mail so a cleaver Mac user made
SpeedMail that will 'reindex' saved email by 'vacuuming' (archiving in Windows terms) saved email. One warning though about 'vacuuming Mail' IMHO don't have an email setting in your INBOX when using it. If the email is still not on the server, the INBOX email might disappear during the vacuum.
Lastly while IMHO a user will need to 'Vacuum' their OS X Mail about once a year especially if they save years and years worth of email! This goes for Exchange Users also to 'archive' about once a year in their Outlook accounts.