I have had this issue a few times. I have a GMAIL account and I would like to b able to check mail from other addresses on it. I successfully managed to do so with a couple of accounts. However, I am still struggling to link others to it. I keep getting the following error message:

Server denied POP3 access for the given username and password

I have had this issue with a couple of gmail accounts. Please advise how to fix this problem!Sounds like you want to have more than one Gmail account. Is that right? On my system I have for email accounts with email and I am able to access them WITHOUT the need to switch my user account.
There are a number of different possibilities you can have. Multiple operating systems, several WINDOWS user accounts, different email clients and an email SERVICE that is browser driven.
From your description I'm not able to clearly grasp what combination of things you're using that does not work for you.
Here is an experiment you might try that to should not cause any harm. Create a new user account on your Windows computer and then LOG into your favorite browser and to yourself to the Google online email account. Offhand I don't remember exactly what the URL is, but I think if you just Google Gmail it will give you a link into email browser-based client. When you sign in and use your username and password, you should see somewhere an option to switch to another account. Are you not able to find that online version of Google Gmaill? Please clarify.