She can't sing AND/ OR dance

From CoGEL p. 934 § 13.31, And in relation to or

Because and and or contrast with one another in meaning, or following a negative is in some respects equivalent to and. Thus:

• He doesn't have long hair or wear jeans. [1]

is logically equivalent to the combination of two negative statements

• 'He doesn't have long hair AND He doesn't wear jeans'. [1b]

This equivalence follows strictly from the so called De Morgan's law of mathematical logic.

• [(NOT (A or B)] ↔ [(NOT A) and (NOT B)] (equivalence, symbolized by "↔")

Given that no ellipsis is used, in other words, given that no distributivity in reverse is applied, the mathematical law is strictly valid in language (any language); I mean by that that "He doesn't have long hair AND He doesn't wear jeans" is not taken as identical to "He doesn't have long hair AND wear jeans".

§ 13.31 goes on as follows (without too much detail, which I'll try to supply)

Conversely:

• He doesn't (both) have long hair and wear jeans. [2]

is logically equivalent to the inclusive disjunction of two negative statements:

• 'EITHER He doesn't have long hair OR He doesn't wear jeans (or both)'. [2b]

This equivalence follows again strictly from the De Morgan's law of mathematical logic, which is true for "and" as well as "or".

• [(NOT (A and B)] ↔ [(NOT A) or (NOT B)] (equivalence, symbolized by "↔")

Thus, [2] (He doesn't (both) have long hair and wear jeans.), which we take as [1b] (He doesn't have long hair AND He doesn't wear jeans) because of the ellipsis that is matter of fact in language (but foreign to logic) appears to mean both [1] and [2b],

• He doesn't have long hair or wear jeans. [1]

• EITHER He doesn't have long hair OR He doesn't wear jeans (or both)'. [2b]

which is nonsense. The explanation is that the coordinator in [1] and [2] is within the scope of negation; more precisely in [1b] and [2] the linguistic distribution of the negation, which yields equivalent statements ([1b] and [2]), is not valid in logic.

{He doesn't have long hair} AND {He doesn't wear jeans}' [1b]

He doesn't (both) {have long hair and wear jeans}. [2]

§ 13.31 goes on as follows

The reversal of meaning arises because in [1] and [2], the coordinator is within the scope of negation (cf 10.64).

Hence or tends to replace and in contexts which we have called NONASSERTIVE (cf 2.53), and more generally in subordinate positions in the sentence:

Soldiers who mutinied or deserted were punished by death. [= 'Soldiers who mutinied were punished by death, and soldiers who deserted were punished by death.'] If we complain or demand compensation, nothing happens. [= 'If w'e complain nothing happens, and if we demand compensation nothing happens.'] He is good at painting with watercolours or with oil paints.

It is this linguistic feature of taking "He doesn't have long hair AND He doesn't wear jeans" as identical to "He doesn't have long hair AND wear jeans" which induces what CoGEL terms "in some respect an equivalence between or and and" and "the tendency for and to replace or in some non-assertive context" (here essentially contexts involving negation).
This means that "She can't sing or dance." and "She can't sing and dance." can be taken as meaning the same thing.
I believe that "She can't sing and dance." can also mean "She can't do both at the same time". In a context that makes that clear, if the sentence is being used as a reminder or as a repetition of the fact in a shortened form, I think the form is acceptable; otherwise it is ambiguous and shows bad style.

For instance, such a dialog as the following would be acceptable

— No, that's a fact, she can't read or write.
— You're telling me she can't read and write, yet, she is in a class where all children know how to read an write… How do you explain that?

English isn’t strictly logical.

In your example, “can’t” can be read as “is unable to” (like “cats can’t fly”) or “is not supposed to” (like “you can’t walk on the grass”), or even as a constraint on the other speaker (like “you can’t be serious”). In each case, though, the natural parsing for “and” is as a conjunction of the two items. However, this isn’t because “and” follows “can’t”.

Consider the following:

• You can’t come and go as you please.

Coming and going are mutually exclusive, so the intent is to place them in disjunction. The intent is not to prohibit the (empty) conjunction.

• You can’t eat your cake and have it still.

The intent is to disclaim the (simultaneous) conjunction.

• You can’t cook and not eat.

The intent is to object to both, occurring in sequence (eating is done after cooking).

There are other examples. Conjunctions are tricky because they can function in so many ways: sequencing (and = then), grouping (and = together with), and purpose (and = in order to), etc. “Or” and “and” are sometimes interchangeable, sometimes not.

• You can have tea and coffee. (You are allowed to choose one or the other, but it would be odd to serve a mixture of both in one cup.)

Unfortunately, disambiguation might require knowledge of how the specific terms are used.