Only if Sprwt enables it for you. There is a setting called Allow Orders on Past Open Dates (options: Allow only checkout to open dates after today — the default, or Allow Checkout to open dates before today), and with the default, an open date that has already passed is not offered at checkout even though it still shows as open on the dashboard.

In the current version this setting is not shown on your Configuration page — it has no tab assigned, so it only appears on Sprwt's internal Super-Admin config screen (under the "Misc." section) and can't be changed by a merchant directly. If you need it changed, contact Sprwt support. This is independent of the Cut-off Timer, which closes dates ahead of time; use the auto-finish delay to make sure old dates do not linger as open.