In this paper, we investigate quasi-abstractive summaries, a new type of machine-generated summaries that do not use whole sentences, but only fragments from the source. Quasi-abstractive summaries aim at bridging the gap between human-written abstracts and extractive summaries. We present an approach that learns how to identify sets of sentences, where each set contains fragments that can be used to produce one sentence in the abstract; and then uses these sets to produce the abstract itself. Our experiments show very promising results. Importantly, we obtain our best results when the summary generation is anchored by the most salient Noun Phrases predicted from the text to be summarized.