When you find yourself using the phrase and yet, consider whether any meaning would be lost if and were dropped. When yet is used as a conjunction, and yet is redundant, and and could usually be cut. For example, and serves no purpose in this sentence:
The numbers do offer a sobering picture, and yet it’s far from all gloom and doom.
And yet is commonly used to start sentences. In some cases, the usage comes from unfounded bias against using yet to start a sentence.
And yet no one would bet against Jobs being on this list in 10 years’ time. [Independent]
Here, there would be nothing wrong with,
Yet no one would bet against Jobs being on the list in 10 years’ time.
Elsewhere, and yet at the start of a sentence is rhetorical shorthand, often followed by a comma and usually meaning, “That may be so, but…”
Ms. Hill, a 50-year-old voice-over actress, said she had been feeling a spiritual drift away from Christmas for several years. And yet, each December she continued to go through the motions of sending out holiday cards, decorating the house, buying gifts. [New York Times]
This use of and yet is less questionable because it’s a common figure of speech.
And yet is not always redundant. When yet is an adverb, and yet works fine—for example:
It is also expected that tax rises and spending cuts, both those already announced and yet to come, will weigh heavily on the economy. [The Sunday Times]