Yes, technically.
The company offering the Big Deal agrees to an expiration date (usually 6 or 12 months). After that time they are not required to honor or refund the promotional offer.
During that time, so long as no work has been performed, the client can request a refund directly from Angie's List.
I recommend contacting the Big Deal provider and asking if they can either still honor the deal or if they would volunteer to replace it with another more current offer.
As an online interior design firm that has sold over 500 big deals through Angie's List, we want to create long term customer relationships - not try to get out of having to provide services for an expired deal.
Source: http://hmdhome.com