I am in total agreement with GavestonPS in his assessment of COA. Interestingly, I hated the show the first time I saw it. I friend of mine told me that he loved the score, so I purchased the OCR, and ended up really liking it, playing it frequently for years, I would imagine (not so much these days, no reflection on its quality). Because I loved the OCR, I decided to see the show again before it closed. I hated it again. I just thought it was very clever for about 10 minutes, then I got bored with the conceit. I could not wait for it to end; even the musical numbers did little for me, I assume because the were preceded and were followed by boring dialogue scenes. I have always felt that my opinion was validated by the general public, insofar as it only ran for 600+ performances, despite winning the Tony, and was discounted for much of its run. I do not know, but I always assumed that it was a financial failure, again despite being a big Tony winning musical.
Here is where I deviate from Gav. As staged by Tommy Tune, with incredible music and lyrics by Maury Yeston, I thought the original production of Nine was a borderline masterpiece. There were so many show-stopping numbers, that someone not there would find that hard to believe. The visual images were striking and the very last moment, a white tiled stage with aquamarine lighting intending to evoke a Venice water everyone wishes were that color (maybe it was intended to be the Lido -- don't remember), and the releasing of doves, with the lush Maury Yeston music being played by the orchestra, represents one of the most memorable stage images in my lifetime of theatergoing. Alas, it was also not a big hit, although it ran longer than COA and I believe returned its investment; I always assumed it was a little too arty for the average musical theatre going audience.
The other issue that originally kept it from being a masterpiece for me was the Guido character. Guido was very similar to Bobby on Company in that he is frequently an observer, rather than an active participant. I was never much of a Raul Julia fan, so I am not sure whether it was the writing of his character or his performance that hurt the overall production. Based on the performance that Antonio Banderas gave 15or so years later, I eventually concluded that it was his performance, as Banderas' performance led to one of the few times ever when I teared up at a theatrical performance. Based on that production, however, which I did like, I did conclude that it was the combination of Yeston and Tune that made that for me a great evening in the theatre.