As long as you've got plenty of processing speed and memory, plus a fast internet connection, you should be able to do multiple things at one time.. The new dual core processors perform this type of activity quite well..
If you start playing resource intensive games, you may start getting some choppy play simply because the large amount of data can't be digested fast enough...but games such as solitaire or chess shouldn't be a problem.
I'm not sure why one station would play while another would not.. Do those particular stations have problems during normal access?
Hope this helps.
Grif