Yes, such people are called "Squibs". We first see this in Harry Potter and the Chamber of Secrets, when we are told that Argus Filch - the caretaker of the school - is a Squib.
"And what on earth’s a Squib?" said Harry.
To his surprise, Ron stifled a snigger.
"Well – it’s not funny really – but as it’s Filch ..." he said. "A Squib is someone who was born into a wizarding family but hasn’t got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual. If Filch’s trying to learn magic from a Kwikspell course, I reckon he must be a Squib. It would explain a lot. Like why he hates students so much." Ron gave a satisfied smile. "He’s bitter."
Harry Potter and the Chamber of Secrets, chapter 9: "The Writing on the Wall"
Other examples include Arabella Figg (as we see in The Order of the Pheonix), and Neville Longbottom was thought to be a Squib at first.
"Arabella Doreen Figg," said Mrs. Figg in her quavery voice.
"And who exactly are you?" said Fudge, in a bored and lofty voice.
"I’m a resident of Little Whinging, close to where Harry Potter lives," said Mrs. Figg.
"We have no record of any witch or wizard living in Little Whinging, other than Harry Potter," said Madam Bones at once. "That situation has always been closely monitored, given ... given past events."
"I’m a Squib," said Mrs. Figg. ‘So you wouldn’t have me registered, would you?"
"A Squib, eh?" said Fudge, eyeing her suspiciously. "We’ll be checking that. You’ll leave details of your parentage with my assistant Weasley. Incidentally, can Squibs see Dementors?" he added, looking left and right along the bench.
Harry Potter and the Order of Pheonix, courtesy of Pottermore