Yes, you can do that. I do that with several email addresses.
There are a couple of steps you need to complete:
You need to have those other email addresses either automatically forward your email to your Gmail address, or have Gmail pull the messages via POP3 (
Settings > Accounts > Check mail from other accounts (using POP3): Gmail help).
You will also need to set up Gmail so that you can "send as" those other addresses. That setting is found under
Settings > Accounts > Send mail as: (Gmail help). When you add an email address, Google will send a message to that email address with a link you need to follow to confirm that you "own" that address.
Once confirmed, you can now send new messages or reply to messages and change the "From:" address being used. One very useful feature is to turn on "When replying to a message: Reply from the same address the message was sent to" so that your reply defaults to the same address where the message was originally set. (You can always change it manually, but it is nice to keep from leaking your Gmail address to, say, your co-workers.)