While WebAssign displays due dates and times in your local time, they are determined by the clocks of the WebAssign servers, not by the date and time indicated on your computer. Every effort is made to ensure that these server clocks are accurate.
To avoid this problem, be sure to submit assignments well before the last minute. Note that the time displayed at the top of the page in WebAssign is the WebAssign server time when the page was loaded into your browser and might not reflect the current time on the WebAssign server.