In mathematics genuine rediscovery seems to me quite common, and it is usually plain that the rediscovery is not plagiarism for reasons related to its content and exposition.
Often the rediscovery occurs in a substantially different subfield than the original discovery and is motivated by different considerations than those that motivated the original discovery. In other cases the result is essentially the same, but the techniques used to prove it are different. In other cases the results can be proved to be equivalent, but this is not self-evident from their statements, or to their authors (or others), and it requires some work, or at least some insight, for people to realize that something is being rediscovered. Sometimes a result that is now considered important in and of itself was proved in passing as a supporting lemma in an old paper, and only once the rediscovery has taken place is it recognized that the result had already appeared. Many results are forgotten - at the time they are published there does not exist an audience prepared to understand their significance - later this audience comes to be and the result is rediscovered - still later someone in that audience realizes that the result had been obtained much earlier.
Finally there is what might be the most common pattern - at the time the relevance of the discovery was not recognized or the work of the person who made it was not paid much attention (for various reasons, e.g the worker published little, wrote in a strange language, died early, etc.). Others come along, and with similar motivations pursue the same lines of research. It is usually clear that such rediscovery is not plagiarism simply from the style and organization of the exposition. More often the results are not exactly the same in the so-called rediscovery, although there may be some overlap.
Each of the scenarios described can be illustrated by numerous examples, but as doing so takes some work (to get the details straight) I'll limit myself to the following two that are more or less fresh in my mind.
Here's an example. It is a theorem proved by C. Lebrun in 1987, and generally attributed to Lebrun, that there exists no integrable complex structure on the six-dimensional sphere that is orthogonal with respect to the standard Riemannian metric. The existence (or not), on the six-dimensional sphere, of an integrable complex structure (with no further condition) is a famous open problem in differential geometry (and one for which many false "proofs", both affirmative and negative, have been published, even quite recently). The theorem proved by Lebrun is an important partial result in that it shows that the most natural ways of proving an affirmative resolution fail. Recently the open problem has been revisited and it has been realized that it (actually something more general) was proved in 1953 by A. Blanchard (surely some specialists knew this earlier, but the broader "community" seems not to have (I learned it from a comment made by R. Bryant on MathOverflow); in a different context, a different result from the same paper of Blanchard was rediscovered by A. Sommese, as he explains in the his MathSciNet review MR0397030 of a paper of S.T. Yau). Blanchard's article has probably been overlooked partly because it was written in French and Blanchard only published a few papers, so probably his work was not well known to later workers, particularly as the use of French in mathematics declined; more likely, people weren't ready to appreciate its import it at the time, when the general understanding of integrable complex structures was still undeveloped. (A review of this particular theorem can be found in this article of A. Ferreira.)
Here's another example (really of independent discovery). There is a mathematical structure now known under the names left-symmetric algebra, Vinberg algebra, pre-Lie algebra, and chronological algebra (among others - the multitude of names for one object is already indicative of the sort of lack of communication that often underlies rediscovery and independent discovery). It was discovered essentially simultaneously in three radically different contexts in the early 1960s. E.B. Vinberg introduced such algebras under the name "left-symmetric" in his study of homogeneous convex cones (these are finite-dimensional left-symmetric algebras). M. Gerstenhaber introduced the graded (so slightly more general) version of these structures, under the name "pre-Lie" algebras, in his study of deformations of algebras. They are also closely related to the B-series introduced by J. Butcher in the context of the analysis of Runge-Kutta type methods for the numerical solution of ordinary differential equations (here it is infinite-dimensional pre-Lie algebras that are relevant). The "chronological algebra" terminology arose in the work of A. A. Agrachev and R. V. Gamkrelidze on control theory. These contexts are superficially quite different and the particular algebras that appear in the different contexts have different characters. There was for a long time essentially no overlap between people working in these areas. It took many years for people to assimilate that essentially the same mathematical structure was being studied in parallel in different contexts, with different points of view and different techniques. In retrospect there are good reasons for the appearance of this algebraic structure in these apparently diverse contexts, and understanding why the same structure was appearing in these apparently disparate contexts is itself quite illuminating.
I view these kinds of rediscovery and independent discovery as useful. They often shed new light, illuminate complementary points of view, or bolster confidence in complicated arguments. It is desirable to give credit where credit is due, but the obsession with novelty for its own sake in publication sometimes ignores that methods and exposition can be novel even if the results are not, and that this can be useful for better understanding and utilizing the results.