In his essay “Politics and the English Language,” George Orwell famously mocked the use of not un- in English. His extreme example is, “A not unblack dog was chasing a not unsmall rabbit across a not ungreen field.” While this sentence is ridiculous, it’s also a straw man. Orwell and other English usage authorities are justified in disparaging the use of not un- constructions in general, but there are exceptions, and not un- can be useful for conveying shades of meaning that positive constructions don’t get across.
For example, some not un- constructions are useful for damning something with faint praise:
Effort’s gentleness and modesty are not unappealing, but these qualities simultaneously provide the picture with an underwhelming impact. [Variety]
As the writer of this sentence doesn’t like the film, he is purposely avoiding positive adjectives so as not to give the wrong impression.
And sometimes not un- constructions describe a middle ground for which neither the positive nor the negative term would be accurate—for example:
London’s buses are not unsuccessful, but they are now somehow deeply unlikeable. [London Evening Standard]
At 80 minutes, the set seemed short but not unsatisfying. [Minneapolis Star Tribune]
Its odor, musty but not unpleasant, is unlike anything else in the city. [New York Times]