Yes, you can put an offer in on as many homes as you like... you only need to perform on any one offer. A good agent will know how to guide you through the process if you get more than one offer accepted.
when an offer is made on a property that is not a short sale, the seller makes decisions based on the indication that you are a serious buyer. if you were my client, i would be very hesitant to make a second offer, unless you are indeed, ready, willing and able to acquire both properties
It is possible to put two offers in on two different homes. However, legally you should be able to afford both homes before you do this. The reason is, you could get both accepted at the same time and be under contract, or legally binded, to carry out both of the contracts. There are ways to go about this that I can help you with strategy-wise. Contact me if you have any other questions regarding this.
i wouldn't recommend doing this unless you put them in on a short sales. If you were doing just a normal offer, if you were my client, i would recommend that you don't do that unless you can qualify for both and am in a position to buy both if it comes up.