你看的资料的说法显然是错误的。语料库有大量使用not both的例句。

You can go there by train or by bus, but not both.

You can take an apple or an orange, but not both.

补充

我注意到刘清沅老师引用《剑桥英语语法》第375页的内容。该书在这里讨论限定词all和both的修饰问题，没有涉及作为代词和副词的 all 和 both。

该书认为限定词 all 可以被 almost 修饰，但 both 不可以。这是可以理解的。因为 all+复数名词可以指很多人或事物。almost all 表示几乎所有的人或事物是符合逻辑的，但 both+复数名词表示二者都，如果说 almost both 表示几乎二者则不符合逻辑。因此，almost 不可以修饰both 是有道理的。但该书对限定词 both 不能被 not 否定的解释则难以令人信服。not both 表示并非二者都，要么只有其中之一，要么二者都不。这并不违反逻辑，为什么不可以说 not both 呢？

以下为 not both+复数名词（即 both 为限定词）的语料库例句：

Students possessing one but not both characteristics are also welcomed onto the Course. BNC

This theme reflects the growing recognition that neither private nor public employers are well-suited to take on the " settlor " or " trustee " roles that the current trust-based framework imposes on them (and certainly not both roles at once).  COCA

In their critical review of the comorbidity of these conditions, Pennington and Bishop (2009) attributed these variations to three factors: (a) the age of children included in the studies, as identification of language impairment declines with age and reading difficulties become more apparent as children progress in school; (b) previous studies of reading difficulty have included children with speech or language difficulties, not both conditions; and (c) clinical samples may exhibit sampling bias that leads to overrepresentation of comorbidity.

Speed has been unchecked, which means the printer prints only in one (not both) directions;... COCA

These findings are partially consistent with the two earlier studies that found that either major depressive disorder (Strakowski et al., 1996) or bipolar disorder (Neighbors et al., 2003) were underdiagnosed, but not both forms of mood disorder. COCA

The good thing about this design is that you can bail out on one side of most of the holes, but not both sides. COCA

The students can use either the duple-meter set of three rhythms or the triple-meter set of three rhythms and the notes from either scale (but not both scales) to compose a song.

This bill has -- it requires one, not both parents. COCA

以上例句not both+复数名词，both显然是限定词。但可能not不是直接否定both，而是省略句，not实际否定的是谓语动词或其它词项。

至于both作为代词作名词词组中心词，以及both...and...前面被not否定的例句太多（280多条），就不一一列举了。

因此，至少代词both和关联连词both..and是可以被not否定的。我们不能一概而论地说not both这种形式是语法错误。副词both被not否定也是很常见的。例如：

They are not both doctors. 他们二人并非都是医生。