Every product have to have it's own unique SKU/model name. That is necessary to track the products inventory. If your product have any attributes, you have to give the product attribute a name. For example : a shoe store. A shoe comes in different sizes and colors. So on the product page, tab attributes options select a default option (blue and size 40 & 41) and the other options that are available for this particular product. Then on the adjustments tab you name the selected options. They can't have the same name/model. For example: NAMESHOE BLUE 41. then it tracks that shoe with color blue and that size.

Also check the access rules. Maybe customers can't purchase more then 1 item or so.
Hopefully this helps, good luck.

Oh and also if the SKU have a ' or a " in it (different then letters) you will have to change some code. Go to http://www.ubercart.org/contrib/132 to check it out. (second comment: minor bug)

Thanks for trying to help. I didn't have attributes on since they seem to have to do with stores where they have different instances of the same thing. My client sells unique items, vintage clothing items. But I went ahead and defined an attribute of size and set a default of medium and set my test item to that. This seemed to me to set up a weird situation where I have two sku/model numbers for the same single item, one sku in the main item screen and the other for the medium sized version. They are one and the same item, of course, which is why I didn't think I needed attributes. Anyway, if I tell it I have just one of those items, and then buy it, after the sale is complete, the item is still there, and there is still a stock of one. Any suggestions or diagnostics? Thanks.

See below about attributes. But I do have unique skus and the problem isn't the customer can't purchase more than 1. The problem is that there is only one of the item and the system will sell it over and over.