I tried two things.  First I enabled multiple signins for both and then I logged out both accounts, and then tried logging in different orders.

 

YouTube then Google Apps

I logged into my YouTube account first (the one I'm using now, exa...@gmail), then l opened another tab and logged into my google apps account provided by the college I goto (exa...@g.rit.edu), no conflicts after I enabled multiple signins for both of them.  This worked fine.

 

Then I cleared login for both of them.

 

GoogleApps then YouTube

I signed into my google apps provided by the college I goto first this time.  Then when I tried to sign into YouTube, it didn't take me to the enter username/password page, but rather a page that says create an account or something along the lines, and there was nothing I could do to sign into YouTube until I closed the browser because logging out of my account still took me to the create account page.

 

So it looks like you can sign into multiple accounts if you do it an external service first (i.e. YouTube) and then login into another Google Apps account.  Ofcourse, my case may be a little different since I'm using two different emails, but it wouldn't let me have multiple signins until I enabled it for both of them.