This may not bear on your question directly, but as a reminder: The choices you make in the Color Settings dialogs are your preferred color spaces. This does not ensure that you're actually editing in that color space. Since the default policy is to preserve the color profile of an image you're opening, the color space you're actually editing in can be defined by the image, or by the method (e.g., Camera Raw) that you open the image.
Are you just opening up untagged images? Where are you getting the images from? If you open up an untagged, unconverted image and you convert it to sRGB ( use relative colormetric) then you pushed all the out of gamut colors into the sRGB space. If you want to keep the larger gamut of colors convert to Adobe 1998 space which is somewhat larger.
I'm tying to understand what colors that I put on web pages look like on users computers, if I use colors that are inside the sRGB gamut vs. colors that are outside of the sRGB gamut. When I am creating graphics in the sRGB color space, I get the out of gamut warning in the Photoshop Color Picker way before I hit the saturation limit in the Color Picker.
I assume that most users are not using color managed browsers, so the sRGB color space is assumed, and that colors that are within the sRGB gamut look fairly consistant on different computers, but on my calibrated HP w2207 LCD monitor I can go way beyond the sRGB gamut. I assume most other users LCD displays can also go way past the sRGB gamut.
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It may help your understanding to know the lay of the land a bit better...
Most users are using Internet Explorer, which is HALF color-managed and reads the document color profile - but assumes the monitor is sRGB, so that anyone using IE with a monitor that's reasonably close to sRGB will see reasonable colors and those who have wide-gamut monitors will always see inaccurate (oversaturated) colors.
Safari and Firefox do use both the document and monitor profiles, and assuming the profile actually DOES describe the monitor performance (which I htink is a stretch for the majority of systems) can show proper color. Firefox has some advanced settings to allow a user to set it to a very rational strategy, in which not only images but web page elements are also color-managed, but that's not the default.
I don't know about Chrome's color-management capabilities. I'm generally allergic to running Google malware software so I don't test it much.
But know that color profiles are complex beasts, and can actually be (mis)crafted so that some or all color-management doesn't actually work right.
Gary Ballard (aka gator_soup) recommends publishing web images without a color profile, claiming this can yield a greater likelihood that people will see your images as intended, but I'm not sure I buy into that argument. Personally I lean toward getting the color right in Photoshop, publishing with an embedded sRGB profile, and not worrying about it any further. Anyone who cares about color will work things out on their own system so they're seeing proper color, and as for the others, well, as long as red things are vaguely reddish, and blue things bluish, what's to worry about?
At the end of the day, there are probably more "standard gamut" monitors (i.e., not wide gamut) out there than anything else, so the likelihood that an image will be seen on a device that's sRGB-like is probably higher than all else. This may be why Microsoft feels comfortable with leaving IE broken as it is, though frankly the same thing would be accomplished if they used the monitor profile, since Windows defaults to associating sRGB IEC61966-2.1 with monitors.
If your prime goal is web publishing, consider just making sRGB your working profile of choice, then watch your histograms when you edit to ensure you don't clip black points of any of the channels. Working in sRGB you cannot have an out-of-sRGB gamut color, so that's a non-issue.
By the way, you can show (as a status) what the current color space of your document is in a couple of ways: The little status box at the lower-left of the document editing area, or via the Info Panel.
Thanks for the thoughtful reply.
I have been using the sRBG profile to create graphics but do not embed the profile in the image file, since it would make the file larger and it is the default profile anyway.
You said: "Working in sRGB you cannot have an out-of-sRGB gamut color, so that's a non-issue."
That was my understanding, as I said in my initial post: "When I have sRGB selected as my color space in Photoshop (CS6), often I see the out of gamut warning in the color picker. I expected every possible sRGB color value to be within the sRGB color gamut, but apparently I was wrong."
Why does the out of gamut warning appear with hightly saturated colors in the color picker? I am using the sRGB color space.
I'll try to make a crude sRGB colorspace map identifying where the out of gamut warning appears.
The out of gamut warning in the Color Picker is telling you that a colour will be out of gamut when converted to your CMYK working space. Ignore it if your not intending to convert to CMYK.
I have returned to the opinion that all color values are within the color gamut of a color space.
I noticed that the Gamut Warning under Views is grayed out when the sRGB color space is selected (see image). The gamut warning in the color picker still appears when the sRGB color space is selected however. Apparently it is indicating the sRGB color is out of the Working CMYK color gamut as Conroy said, not the Proof Color Space gamut.