How about this phrase Не вижу смысла (It doesn’t make sense to me).

The verb видеть is transitive, so according to the rule the noun смысл must be in accusative. Yet смысла is not accusative, but genitive case (check out the complete declension table).

Besides, if I were to say Я вижу смысл в чём-то (Something makes sense to me), the noun does conform to the rule and is in accusative.

Why would you say

Я понял вопрос – I understood the question

but

Я не понял вопроса – I did not understand the question

and

Я помню твой адрес – I remember your address

but

Я не помню твоего адреса – I do not remember your address

and

В его произведении есть глубокий смысл – There’s deep sense in his work

but

В его произведении нет никакого смысла – There’s absolutely no sense in his work

The only difference between each pair of sentences is that the first sentence is a positive one and the second is a negative one.

So it looks like even if a verb is transitive, but is used in a negative sentence, genitive case will be used for a noun.

That is indeed the case except… ok, let’s consider the following examples:

Я не купил эту книгу (I did not buy this book) – The word книгу is accusative of книга (book).

Я не купил его преданности (I did not buy his loyalty) – The word преданности (loyalty) is genitive of преданность (loyalty).

Не урони тарелку! (Do not drop the plate!) – the word тарелку is accusative of тарелка (plate)

Не урони достоинства! (Do not demean yourself!) – the word достоинства is genitive of достоинство (dignity)

Сам я не вынесу стол (By myself I won’t carry the table out) – стол is accusative of стол (table).

Я не вынесу этих трудностей (I won’t endure these difficulties) – трудностей is genitive of трудности (difficulties)

You’ve probably noticed that accusative was used for tangible nouns – a book, a plate, a table. Genitive was used for abstract nouns, such as loyalty, dignity, and difficulties.

However, this is not a hard and fast rule, but more of a general guide. We prefer to use accusative for tangible and genitive for abstract nouns. But we don’t always do. That is why you are as likely to encounter both

Я не понял вопроса and Я не понял вопрос (I did not understand the question)
Мы не достали билетов and Мы не достали билеты (We did not get the tickets)
Он не уронил тарелки and Он не уронил тарелку (He did not drop the plate)
Ты не вынес стола and Ты не вынес стол (You did not take out the table)
Она не съела борща and Она не съела борщ (She did not eat borscht)

To confuse the situation a bit more, idiomatic expressions are excluded for this altogether and the correct declension must be memorized:

Не морочь мне голову (Do not pester me)
Не заговаривай зубы (Do not give a runaround)
Не находить себе места (To feel antsy)
and more

If you feel confused, take solace in the fact that many native speakers tend to just as confused about the use of accusative and genitive cases in negative sentences. Remember, the more you speak, the higher the chance you will get it right!