Is it correct that "etc." can not be used together in a sentence with "for example" and "such as"?

Apart from being redundant, as Mitch said in the comment, I don't think etc. fits very well in that example. There is a nuance to the use of etc. that Wikipedia sums up:

The phrase et cetera is often used to denote the logical continuation of some sort of series of descriptions

"Logical continuation" is the key here. Consider these two sentences:

A balanced diet should include fruits: apples, oranges, etc.

The fruit basket contained some of Bob's favorite fruits, for example, apples, oranges, etc.

In the first example, you can logically continue the list of examples by substituting any type of fruit. But in the second, you can't. You have no way to know what other types of fruits Bob likes to eat, nor what was in the fruit basket.

I don't think it's necessarily wrong to use it in that way, but it does feel very awkward to me.

Here is a reference. In their The Elements of Style, Strunk and White discourage the use of "etc." at the end of such lists.

Update: Another reference is A Handbook for Scholars (2nd edition) by Mary-Claire Van Leunen. On page 130 she writes: "For example" gives notice that only some members of a set are to be mentioned; "etc." then drags in all the other members. The two expressions are at loggerheads.

Merriam-Webster's Dictionary of English Usage, p. 412, notes:

A few commentators warn that etc. should not be used at the end of a list introduced by for example or such as (as in “...such photographic materials as lenses, filters, etc.”) The redundancy of etc. in such contexts is not strongly felt, however, and the usage considered erroneous by the commentators does occur in standard writing.

Merriam-Webster's position is that since this usage occurs in standard (meaning professionally produced and professionally edited) writing, they have no objections.