Which makes more sense in American English?

The non-restrictive relative clause:

The bed has a thickness, which may be adjustable.

versus the restrictive relative clause:

The bed has a thickness that may be adjustable.

The use of the word may suggests to me that it is non-restrictive, however, I feel like it is more correctly written as a restrictive clause. Are both acceptable, and if so, how do their meanings differ? The meaning I desire, is a bed with a thickness, that may or may not be adjustable (i.e., the bed's adjustability is not determined).

These sentences don't make much sense, that is because I took a much longer sentence and both genericized it and stripped it down to the the portion that applied to my question. I think the key point is if the relative clause contains words like may or could that make the clause optional, does it make sense to ever use the restrictive clause?

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    I wouldn't use either one. Don't all beds have a thickness? So why bother saying that it has one? How about: The bed's thickness may or may not be adjustable. – Peter Shor Oct 30 '15 at 19:05
  • This is one of those cases where it doesn't make a lot of difference. The only real distinction is the independence of the thickness and its adjustability, which is not relevant to anything real. And since it's short, there may or may not be a phrase break justifying non-restriction. – John Lawler Oct 30 '15 at 19:07
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    The bed has adjustable thickness. Brevity is the merchant's best friend. – Ricky Oct 30 '15 at 19:09

The second one (restrictive) more correctly communicates your meaning. All mattresses have "a thickness." So there is nothing about that sentence that provides value without the addition of the subsequent clause. The sentence "The mattress has thickness" is grammatically correct but semantically awkward. That is why the first example ("The mattress has thickness, which may be adjustable") sounds incorrect to your ear.

Compare this to:

  • The pot has a lid, which may be scratched.
  • The pot has a lid that may be scratched.

Both of these "sound okay" to my ear, because pots may or may not have lids -- but the shade of meaning is different (just as it's different in your example). In the first, you are definitively stating the presence of the pot's lid, and then subsequently mentioning the scratch as an "oh-by-the-way." In the second sentence, you are mentioning the lid only because you want to specifically talk about how it might be scratched.

Since the possibility of being adjusted is fundamental to your sentence's meaning, a restrictive clause is more appropriate.

Edit: In response to your edit, in which you ask about the impact of "may" on any general restrictive clause, I would say that it doesn't have any impact. What matters is whether that clause is intrinsic to the quality described or not. The questionable adjustability of the mattress is either a necessary descriptor of the thickness or it's merely additional interesting information. The fact that the adjustability happens to be questionable ("may") is irrelevant.

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