Yes, the system will send text notifications as soon as a site becomes available which may mean the notification comes in the middle of the night. However, this is unlikely as most sites are not cancelled in the middle of the night.
If you do not want to be interrupted by incoming texts during the night, you can place your phone in Sleep Mode:
iPhones: [ link to Apple Support Doc ]
Android: [ link to Google's Android Doc ]
Be sure to also setup Do Not Disturb as part of your Sleep Mode.