First of all you need to understand that the size of a cabinet is the overall dimension of the face frame and thusly, the interior dimension is reduced by up to 2 inches.
Also the clips are not necessarily the best method of installing an undermount and in some material is discouraged.
There are mounting brackets which are designed specifically for that application and are installed using the cabinet as the support structure.

In some cases the "minimum" cabinet size reflects what is considered by some as the sink cabinet being wasted space and not saying that the cabinet may need to be amended to affect a fit.

You'll notice that all of the dimensions are divisible by 3. That is because of an "industry standard" for "bag-n-box" cabinetry, as opposed to custom cabinets.

Hi snoonyb, thanks for responding.

First of all you need to understand that the size of a cabinet is the overall dimension of the face frame and thusly, the interior dimension is reduced by up to 2 inches.

Thanks! We're having frameless cabinets, with sides 3/4 inch thick (2 sides x 3/4 inch = 1.5 inch loss to sides). So, from a 33 inch cabinet, that leaves 31.5 inches in the interior; and from a 36 inch cabinet, that leaves 34.5 inches, right?

I'm confused about what you mean by this sentence:
In some cases the "minimum" cabinet size reflects what is considered by some as the sink cabinet being wasted space and not saying that the cabinet may need to be amended to affect a fit.

The "minimums" in my original post are cited from the manufacturer's pages about their sinks. So, I was assuming that what they are saying is how I buy clothes: if I (the sink) am a certain size, and I'm looking for a dress, the dress had better be that size or bigger. Or else, I won't fit inside of it. :-)

So, were you indicating that when a sink manufacturer says the minimum fit needed is XYZ, that that is only a guideline of some sort, and has something to do with their thoughts of the cabinet interior as wasted space?

I'll rewrite this: "In some cases the "minimum" cabinet size reflects what is considered by some as the sink cabinet being wasted space and not saying that the cabinet may need to be amended to affect a fit" in another way:
= = Usually, the size declared as "minimum" cabinet size is the safest size to give to the public because it involves no amending to make a fit. In reality, it wastes a fair amount of space when one adopts a strict optimizer point of view.

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Yes, in frameless boxes a 33 inch cabinet leaves 31.5 inches in the interior; and from a 36 inch cabinet leaves 34.5 inches. The amount of space available inside is 1.5" less than the width of the exterior.

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If your sink is greater than these dimensions (31.5" or 34.5") by an inch or a bit more, you MAY have to do some work cutting some of the chipboard out, to allow the sink rim to rest directly on the chipboard sides and also not be too high. OR' you could do no cutting, and instead add a strip of metal filler on top of the chipboard sides everywhere, to raise the level of the counter. I did this and I screwed the metal strips to the chipboard sides, thereby making a solid frame of all the cabinet boxes all screwed together, and also giving the counter a base to rest on that was better than resting it on the chipboard sides. Ta da!

Oh davidro1, thank you, thank you, thank you!!

For your words going to the heart of my question:
Usually, the size declared as "minimum" cabinet size is the safest size to give to the public because it involves no amending to make a fit. In reality, it wastes a fair amount of space when one adopts a strict optimizer point of view.

That is precisely what I was wondering about. Thank you for that.

OR' you could do no cutting, and instead add a strip of metal filler on top of the chipboard sides everywhere, to raise the level of the counter. I did this and I screwed the metal strips to the chipboard sides, thereby making a solid frame of all the cabinet boxes all screwed together,

That is a really neat idea! Adds some structural strength while addressing sink fit. Thanks for sharing that--it never would have occurred to me.

Do you see any downsides to adding the metal filler (other than material cost and the labor)? I mean, I know you said you did it--I'm just wondering if there was a downside that I'm not seeing, which you were willing to live with to get the benefits.

Thanks again for realizing the heart of my question better than I knew how to ask it!

--Lee