Current language models can generate high-quality text. Are they simply copying text they have seen before, or have they learned generalizable linguistic abstractions? To tease apart these possibilities, we introduce RAVEN, a suite of analyses for assessing the novelty of generated text, focusing on sequential structure (n-grams) and syntactic structure. We apply these analyses to four neural language models (an LSTM, a Transformer, Transformer-XL, and GPT-2). For local structure - e.g., individual dependencies - model-generated text is substantially less novel than our baseline of human-generated text from each model's test set. For larger-scale structure - e.g., overall sentence structure - model-generated text is as novel or even more novel than the human-generated baseline, but models still sometimes copy substantially, in some cases duplicating passages over 1,000 words long from the training set. We also perform extensive manual analysis showing that GPT-2's novel text is usually well-formed morphologically and syntactically but has reasonably frequent semantic issues (e.g., being self-contradictory).