Successive relative clauses

Is the use of two or more successive relative clauses common or grammarically accepted in English? As in "The man who is sitting in the wheelchair and who has a broken leg. Or "The man who is sitting in the wheelchair and whose leg is broken."
I know, of course, there are ways to shorten them, but I just want to know if it sounds grammatical to native speakers of English to use successive relative clauses, whether in spoken or written language.

Yes, it is perfectly grammatical. And yes, it is perfectly common, too. Here are just a few of the many examples from the Corpus of Contemporary American English and the British National Corpus, showcasing a variety of patterns (emphasis mine):

• In a more elegiac vein, they read like an homage to the late Cy Twombly, who died this year and whose deceptively elegant scribbles, mysterious pentimenti and cryptic, barely legible text passages are channeled here.
  Houston Chronicle, 2011

• I wasn't actually invited to the party in question, but I had a friend who got me in, and whose name I will not reveal.
  The fourth wall, Williams, Walter Jon., 2012

• It was that young couple whose vehicle would be crushed and whose survival would amaze a community.
  Denver Post, 2012

• The position is, of course, quite different if the witness knows the person concerned well enough to say: 'I was hit by a man I know well and whose name is Jack Spratt.'
  The Criminal Law Review (1985–1994)

• It is outrageous that the name of a man who is so completely discredited, and whose actions run so contrary to everything that America is supposed to be about, should be emblazoned in gold on the federal law enforcement building in Washington.
  Bookseller, London: J Whitaker & sons, 1993.

Edit: in response to comments, here are more patterns still, with additional focus on those most similar to your example, and beginning with one that is exactly the same in every respect.

• "I don't see that it's in our community's best interest to swear allegiance to one Democratic candidate when there is another candidate who is wooing us and whose record is very credible on our issues." Atlanta Journal Constitution, 1999

• In a 1988 case, an employer confronted a worker who'd been performing poorly and whose wife had expressed concern about his drinking, only to be hit with $100,000 punitive damages when it turned out his problem was depression rather than booze. Washington Monthly, 1997

• "Trace amounts of a food allergen can cause a severe reaction, and just because you had one reaction one time doesn't mean it will be the same type the next time," said Schwartz, who is allergic to fish and shellfish and whose daughter is severely allergic to peanuts.
  San Francisco Chronicle, 2010

• I attend the dance training session presented by La 2e Porte gauche (Second Door on the Left), a group influenced by Michel Reilhac's Bal Moderne and whose members presented to the public Marie Chouinard's choreography for Orpheus and Euridice. Art Journal, 2011

Follow the links for yourself to browse the hundreds upon hundreds of more examples.

Grammatically there's nothing wrong with it. An example dialog:

Person 1: Do you see that man?
Person 2: What man?
Person 1: The man who is sitting in the wheelchair and whose leg is broken.

However, to me it sounds sort of long-winded and unnecessarily verbose. Try:

Person 1: Do you see that man?
Person 2: What man?
Person 1: The one in the wheelchair, with a broken leg.

Note that "on the wheelchair" is possible, if the chair is tipped over and you are literally sitting on top of it, for example. A man with a broken leg would most likely sit in the wheelchair, not on it.