There's no current rule against starting a sentence with "and" or "but."
Ultimately, writers must, for the time being at least, decide for themselves which side of the debate they are on and either use or reject the construction accordingly. They must also consider whether or not their decision applies equally to the other coordinating conjunctions—for they are covered by the same rule.
There is much debate about the grammatical correctness of starting sentences with ‘and' and ‘but', though both words are now used to do so. Purists argue that the purpose of these coordinating conjunctions is to link related words, phrases or clauses within a sentence, and that their use is valid only when the link is not interrupted by a period. Those who argue that it is acceptable to use ‘and' and ‘but' at the beginning of sentences say that that interpretation is too literal. They often point to the fact that some great writers, including Shakespeare, have used ‘and' and ‘but' at the beginning of sentences, and that up to 40% of the words' usage in newspapers occurs at the beginning of sentences. They also point out that we commonly use ‘and' and ‘but' at the beginning of sentences in conversation.
Whatever view grammarians hold on the use of coordinating conjunctions at the beginning of sentences, they are almost universally opposed to the excessive use of the construction, and suggest it should be used infrequently. They appear to broadly agree that the excessive use of ‘and' or ‘but' at the beginning of sentences results in writing that is dull and monotonous
