The lawsuit reported on by ABC News did happen (Robert Jordan vs. the City of New London, CT). The court case can be found here. That page verifies many of the facts given in the article, including the conclusion: it was ruled constitutional to not hire someone as a police officer because they scored too highly on an intelligence test (The test used in this case was the Wonderlic Test).
The LA Times in 1997 reported on the story (before it went to court but after the lawsuit was filed). The article, which seems to be heavily based off interviews, seems to be reliable (and some of it is also backed up by information in the court case). It does appear to be the policy of this particular police department to reject people who score too high:
Two other applicants who scored even higher [than Robert Jordan] were also rejected.
Turnover is the reason given for the decision, due to its high cost:
Deputy Police Chief William C. Gavitt and the city's attorney, Ralph J. Monaco, said candidates who score too high could get bored with police work and leave not long after undergoing academy training that costs about $25,000.
"We are looking for bright people," Monaco said, "but we're not looking for people that are so bright to an extent that they're not going to be challenged by the job."
According to the makers of the test used, other places also have this policy:
Wonderlic said New London is not alone in screening out potential employees deemed too smart, but it wouldn't identify any of those employers. If there are any, they are not exactly coming forward to say they don't hire smart people.
However, this certainly isn't the policy everywhere. In particular, it isn't the policy another PD over:
In contrast, the neighboring community of Groton uses the exam only to screen out those who test low in intelligence.
"I go for the highest score on the Wonderlic that I can get," said Police Chief Wilfred Blanchette Jr. "My instructions are, 'You give me a list of people who are above this number.' Let me figure out if they're going to get bored or not."
Scoring too high may also hurt people in other fields. In this case, the field was a football field:
The only perfect [Wonderlic] score belongs to Pat McInally, a former Pro Bowl punter and receiver for the Cincinnati Bengals. Another Harvard man, who might have scared teams with his score.
"In those days, I was told it did hurt me," McInally said from California, where he is a high school coach. "(Giants general manager) George Young pulled me aside at a party in New York and said, 'That cost you a couple of rounds.'
Harvard's Ryan Fitzpatrick gets passing grades for 3-0 Bills