The grammar is fine, but the structure is weak; that is why it does not sound the best. 'Growing', however, is the usual collocation: The opinion that computers are not that dependable is growing. The opinion is growing that computers are not that dependable.
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Mister MicawberThese do not mean quite the same as 'have begun to think', however.Not exactly the same. OK. Then what do you think the difference is?
Mister MicawberGrowth comes after inception.So the 'many people have begun to think...' example doesn't have the sense of growth.