It isn't a silly question, but you haven't given enough information to get more than a really, really general answer. Are we talking Sassie, Prophet, ShopMetrics, isecretshop? Or one of the companies that has its own system, like Market Force? What company are you asking about?
MOST companies, yes, but not all. Usually a shop is offered between 8/20 and 8/24 and you pick a date. Sometimes, when assigned, your shop log (if it's Sassie) will show 8/24. In that case, NO, only the assigned date. Sometimes it will show 8/20-24 on the Shop Log. In that case, yes, for most MSCs. However, I have read recent posts from shoppers who showed a date range on their Shop Log but their shops were disallowed because they did the shop early. It just depends on the company and on how they are set up. If you have doubts, it's a good idea to contact the scheduler or stick to the due date.