While I don't doubt that the increased ease and frequency of piracy has had an impact on the bottom line of the recording industry, I believe the decline of the industry as a whole has much more to do with the increased capabilities of home computers and the spread of broadband Internet access.
The big recording companies provide a few main services to artists: production, distribution, and promotion. For years they kept CD prices artificially high because there weren't alternatives for most people (for artists to create albums or for consumers to obtain music).
Once everyone could record and burn CDs expensive production studios became much less valuable, and broadband Internet makes distribution nearly free (relatively, at least). Simultaneously, radio has become a much less important means of promotion, so the connections that the recording companies had in that arena become much less valuable.
In other words, the big record companies just don't offer the value they used to - either to artists or to consumers. And once the value your company offers disappears, so should your company (unfortunately this usually happens very slowly, with the company(/ies) trying everything to avoid extinction).
So was it a coincidence that this all happened at around the time that Napster took off? I don't think so, I think there were just common causes that led to both results.