The use of both where neither would be clearer and more idiomatic is fairly common, but inherently ambiguous.
Take for example, “Both men were not arrested.” Is the intended meaning “neither man was arrested” or “one man was arrested, but the other wasn’t”?
Here are some examples of statements that use both where neither is the better choice:
Original: Both of us didn’t have too good of a year (in 2013), and we’re seeing that in the attendance.
Better : Neither of us had too good of a year (in 2013), and we’re seeing that in the attendance.
Original: Both of us didn’t want to go to the Hospital but the fireman told us it would be best to get checked out.
Better : Neither of us wanted to go to the Hospital but the fireman told us it would be best to get checked out.
Original: Diaz and Felder tend to vote conservatively on social issues, so they both aren’t going to be happy with the abortion plank.
Better : Diaz and Felder tend to vote conservatively on social issues, so neither is going to be happy with the abortion plank.
Original: Trustees Michael Bubba and Eileen Albanese were also sworn in as trustees. They both have never held elected office.
Better : Trustees Michael Bubba and Eileen Albanese were also sworn in as trustees. Neither has ever held elected office.
In a few contexts, both + negative works:
Intimacy can’t happen if both aren’t present in the exchange.
The core similarity between them is that both aren’t so much political parties as they are social movements.
A general rule to keep in mind, however, is to avoid using both with a negative:
Both Smith and Jones are the children of noted politicians.
Neither candidate has ever held public office.
Both and neither on newspapers
… adults whom the scientists recruited were normal weight and generally healthy; others were overweight; and still others both overweight and diabetic. None exercised regularly. The scientists invited … (www.nytimes.com)
… so often, the couple here present themselves as both outsiders in an elite institution and as heirs to it; as people excluded from its narratives but now possessors of … (www.nytimes.com)
… proud of it. They had found a solution that made sense to me. Why should I stuff myself into binary gender terms when neither male nor female seemed to fit? … (www.nytimes.com)
… sharply escalate the administration’s tariffs on Chinese imports — a threat he reiterated on Friday — shows that neither side has yet gone far enough to persuade the other to compromise. Bigger and broader tariffs may be necessary … (www.nytimes.com)