Yes, clear as mud and you’ve got it right.
When you setup the widget, there may be an explicit option to set a refresh rate – you may have the refresh rate set aggressively and could better increase the chances that the widget stays functional by increasing the polling time – I imagine several machines at your school sitting with the home page of your school’s website open, all polling Twitter for the same data.
The long-term solution for this would be to have your IT department create a central place that grabs the “interesting” tweets to your school, then configure of your widgets drink from that pool of tweets instead.
