both...not... 和 not both...，都对，都是部分否定。例如：

Both boys are not here. （both 为限定词用法）

Both of the boys are not here. （both 为代词用法）

Not both boys are here. （both 为限定词用法）

Not both of the boys are here.（both 为代词用法）

后两句比前两句符合习惯。

各位老师请注意：

我们讨论的话题是“部分否定”：“两者不都...”；不讨论“全部否定”：“两者都不...”。

例如：“两个人不都在这里。” 该怎么翻译？

因此，尽量不要涉及无关的内容，以免混淆。

黎老师引用的这些例句：

Neither of them is here. 他们俩都不在这里。(不能说：Both of them are not here.)

例如：他的两个学生都不在那儿。

误：Both his students were not there. 

正：Neither of his students was there.

不太符合本话题，而且容易产生误导。

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。

    这是一个有争议的问题。

   1.认为 both 通常不用于否定句中。

  《牛津英语用法指南》第三版指出：“通常不说 both ... not，而用 neither（参见372）。Neither of them is here. 他们俩都不在这里。(不能说：Both of them are not here.)”

   Collins COBUILD English Usage 认为 both 通常不用在否定句中。例如：他的两个学生都不在那儿。误：Both his students were not there. 正：Neither of his students was there.

    2.认为 not both 表示部分否定。例如：

    You can take either of the two,but not both. 你可以拿这两个当中的任何一个，但不能两个都拿。（摘自《外研社英汉多功能词典》）

    3.认为 both ... not 既可以表示“两个并非都 ......”，也可以表示“两个都不 ......”。例如：

    Both of them are not students.(摘自英国出版的《英语语法大全》英文版)他们两个并非都是大学生。 或：他们两个都不是大学生。（译文摘自赵振才编者的《英语常见问题解答大词典》）

not与both,all, altogether, everyone可以连用表示部分否定，

直接把否定词not放在both/all前面和把not放在后面谓语否定都是属于部分否定都是正确的。

(1)both 的否定式：not…both (或：both… not) "并非两个……都……" 
I don't want both the books. 

我不是两本书都要。
Both (the) windows are not open. 

两扇窗子并不都开着。

Not both children are clever. 

=Both children are not clever.

两个孩子并不都聪明。

(2)all 的否定式：not all…（或：all…not）表示"并非都……"、"不是所有的都……"
Not all men can be masters. 

= All men cannot be masters.

并非人人都能当头头。
Not all bamboo grows tall. 

并非所有的竹子都会长得很高。

Not all birds can fly. 

并不是所有的鸟都会飞。

(3)every…的否定式：not every/not everyone"不是每……都……" 
Not every book is educative. 

=Every book is not educative.

不是每本书都有教育意义的。
Not everyone likes this book. 

并非人人都喜欢这本书。
This flower is not seen everywhere. 

这花并不是随处可见的。

(4)always的否定式："并非总是（并非一直）……" 
He is not always so sad. 

他并不是一直都这样悲伤。

(5)entirely, altogether, completely 和quite 的否定式："不完全……"，"并非完全……" 
The businessman is never to be entirely trusted. 

不可以完全信任商人。
He felt not altogether satisfied. 

他并不完全满意。
I don't agree completely. 

我并不完全同意。
What he did was not quite proper. 

他做的不十分妥当。

We are not altogether interested. 

我们并不完全感兴趣。