Can I switch company after 15 days?

I've got two offers, from companies X and Y. Company X wants me to start on Monday. Company Y still has BGV pending and wants me to start in the second week of August. I would prefer to work for company Y. I intend on signing both letters.

Can I start working at company X and then move onto company Y when they start? Will company Y's BGV show them that I'm already working for another company? I really don't want to be in a position where I'm waiting for company Y after having let go of the offer from company X, but at the same time I really want to work for company Y. I piam a bit worried about the BGV as it has caused issues in the past (worked abroad prior to this and also the drug test puts me in an iffy spot).