This article is from the Anime Music FAQ, by Ru Igarashi with numerous contributions by others.
No, not at all. Copyright laws protect a work for many decades regardless of whether the rights holder ever acted. The rights holder can in fact take action at just about any time on any of many infractions, though there can be some limits on how long ago a violation can be sued for.
The two main ways of losing the rights, or the for the work to enter the "public domain", is a) the passage of time past the duration stipulated by law (many decades), or b) the rights holder explicitly stipulating the the rights have been relinquished into the public domain.
 
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