Would like your views on what I have said below.
What you are seeing is a competition between two of the basic limits of lens optics.
In simplified terms, designers try to capture all the light striking a lens from a photographic subject and bend (focus) that light so that it reaches the camera's sensor at the right position to create an image. No lens is perfect, so while the light travelling through the very center of the lens will always be in focus, as you move out from the center, the focus starts to vary with the distance from the subject. When you increase the f-stop number (i.e. adjust the iris), you create a smaller internal diameter in the lens, cutting out more and more of the light travelling through the outer areas of the lens, the light that tends to be less in focus at anything other than the current lens point of focus. This is the effect that gives you a "depth-of-field" choice - let in more light and you get a brighter picture but less clear focus across the whole depth-of-field, cut down on the light to get broader focus and you need to decrease the shutter speed / increase the ISO to compensate.
The above process works well on large format lens systems on big cameras ( DSLRs and larger). Unfortunately there is a natural limit to how small you can make the iris opening (see philmoz's post about the diffraction limit) due to the physical effects that happen at the edge of the iris. Again, in simple terms, the light passing very close to the edge of the iris bends differently at different wavelengths (colors), causing an additional loss of focus. When the iris opening is very wide, the percentage of light affected is very small and you won't notice it. When the opening is very small, the effect becomes more noticeable.
What all this means from a P&S camera point of view is that there is a usually "sweet spot" in the range of available f-stops where the increase in depth of field balances the diffraction effects from using smaller iris openings.
Unfortunately, there is no perfect set of settings for every camera and every shooting circumstance. The "Auto" mode in P&S cameras tries its best to optimize settings, usually with some success. However, unless you use the various "scene" modes, the less expensive cameras tend to give you little or no control over the settings used.
To help with this issue, peabody & I published a
CHDK script that gives you an Auto mode that lets you define acceptable ranges for your Tv, Av and Sv settings. You get to define the iris "sweet spot" range that way, pick what you consider acceptable noise level in your ISO setting, and select the needed shutter speed for your subject. We put these into a KAP application to give KAP users the choice of what to use but the code is generic enough to be used in almost any shooting circumstance if you want to experiment and not just accept somebody else's opinion about what is needed. As defined above, it can't change the laws of physics - which we fully understand - it just puts you in charge to make your own choices.
Update : as long as I'm creating a monster post, I should probably tag on a basic definition of f-stop. People sometimes assume that the f-stop is the diameter of the lens iris opening. While it is related to the diameter, it is also related to the lens focal length, which is essentially distance from the center of the lens to the camera's sensor. Simply put, f-stops are the ratio of that focal length to the lens iris diameter. It gives you a standard, "unitless" measure of the amount of light that will pass through a give lens. To the average user, the lens focal length is fixed for a given zoom position, so adjusting the f-stop appears to be the same thing as adjusting the iris opening. The key here is that larger lens assemblies in DSLRs have a longer focal length, and so a physically wider opening can have the same f-stop number as a smaller opening in a P&S with a short focal length. And that smaller opening means diffraction becomes an issue for P&S camera lenses long before it affects DSLR camera lenses.
As a side note, all this leads to some confusion when Canon quotes an f-stop range for their P&S cameras that don't actually have an adjustable iris. The numbers they quote are across the zoom ranges as the lens focal length varies with zoom setting and therefore so does the f-stop. For the most part, the lens opening remains constant.
Update : fixed two small spelling "misteaks" in a very long post. Thanks to microfunguy for reading the whole post and pointing them out.