We can certainly *pronounce* it with the contraction (vowel reduced to schwa, dropped H), but we wouldn’t normally *write* that contraction. In American English, we generally only write contractions after pronouns and some words like “where” and “there”. In UK English, they actually write the contracted forms after proper and common nouns (except in formal writing), but at the end of a multiword noun phrase? No. 

Similar rules apply to the genitive apostrophe + "s". We can use it at the end of a multiword phrase in spoken English, but it would be better to use a prepositional phrase with "of" in writing.

U.S. English:

No one / nobody in this office has his* own computer.

People in this office don't usually have their own computers.

*In spoken English, it is very common to hear "they," even though it is not correct.