An overwhelming exception is a logical fallacy similar to a hasty generalization

Hasty generalization

Hasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence essentially making a hasty conclusion without considering all of the variables...

. It is a generalization that is accurate, but comes with one or more qualifications which eliminate so many cases that what remains is much less impressive than the initial statement might have led one to assume.

Examples:

• "All right, but apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, a fresh water system, and public health, what have the Romans ever done for us?" (The attempted implication (fallacious in this case) is that the Romans did nothing for us). This is a quotation from Monty Python
  Monty Python
  Monty Python was a British surreal comedy group who created their influential Monty Python's Flying Circus, a British television comedy sketch show that first aired on the BBC on 5 October 1969. Forty-five episodes were made over four series...
  
  's Life of Brian.
• "Our foreign policy has always helped other countries, except of course when it is against our National Interest..." (The false implication is that our foreign policy always helps other countries).
• "All Americans are useless at foreign languages. Ok, I'll make an exception for those who live in multi-ethnic neighborhoods, have parents who speak a foreign language, are naturally gifted in languages, have lived abroad or who went to a school with a good foreign language program, but the rest are absolutely useless at foreign languages."
• All dogs are black, except for those that are not black. (This is also a tautology
  Tautology (rhetoric)
  Tautology is an unnecessary or unessential repetition of meaning, using different and dissimilar words that effectively say the same thing...
  
  , or rather a validity.)

See also faulty generalization

Faulty generalization

A fallacy of defective induction reaches a conclusion from weak premises. Unlike fallacies of relevance, in fallacies of defective induction, the premises are related to the conclusions yet only weakly buttress the conclusions. A faulty generalization is thus produced...

 for other fallacies involving generalization

Generalization

A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it...

.