When my wife and I purchased our home in 1971, the procedure was simple.
We bid a price and, if the offer was accepted, we provided a cash binder.
So we knew almost immediately whether we were under any obligation to purchase that property.
If our binder wasn't accepted we knew we could then bid on another property.
Now (in 2007) we are trying to buy a CoOp apartment for our son in New York and we are facing a dilemma.
Today property sellers are no longer dealing with deposits or binders. You make a bid and they either:
1- Ignore the offer and don't get back to you.
2- Tell you the offer is not enough but decline to suggest a counter-offer.
3- Accept immediately (Rare).
4- Tell you they already have an offer of xxx dollars and ask if you can beat it.
5- Tell you the offer is low but would like at least yyy dollars. This of course is called negotiating.
The agent we are dealing with tells us that, once we make a bid, we cannot bid on another property until the first bid is resolved. We have another property in mind as a contigency We have placed our bid and have gone through items 1 and 2 above
Here's my question:
If the seller or their agent is not responding in a timely manner, can we then make a bid on another property? Is there some kind of a structured procedure we can follow?

Set a time for your bid to be accepted or rejected, if no notice by that date, your bid is withdrawn.
That can be anywhere from 3 days to a week.

And you can formally withdraw your bid if they do not get back to you timely

I agree with the Father... put a time limit on your offer. Back when I was buying in California, 24 hours was the time I gave them to consider the offer, to prevent the seller from shopping around for a better offer. In Texas, I was told that a week is more common.