You can place as many offers as you'd like - but you will need to put down earnest money when you make the offer. Just keep in mond that you are "offering" to make a real estate purchase. If you make an offer which is accepted and then bakc out, you will forfeit the deposit - so don't make an offer unless you are prepared to go thru with the purchase it your offer is accepted. The brokers won't usually make the offer to the seller without the deposit/earnest money.