This depends on the quality of your projector. The very best models don't need a pitch-black night to work successfully, but no matter what model you choose, you're always going to struggle if it's a very bright, sunny day. It needs to be at least getting dark before you try to air your chosen film or show, but check with your projector manual for the best advice.
We recommend a lumens of at least 3000 to be able to watch the show in good quality. Most projectors will come with this or higher - the higher the lumens, the higher the price tag is likely to be.