Inbreeding is harmful in a direct correlation to how many recessive genes for illnesses there are in the family.
Longterm, multigenerational inbreeding tends to collect recessive genes and keep them closeted in one segment of the gene pool. If no such genes are present, there's no problem with marrying a cousin.
Mutations tend to happen randomly, so that one could expect that after 800 years or so of inbreeding, that more problems would appear (as compared with the case studies of the 10,000 first cousin marriages, which concluded genetic problems were not statistically much worse than with stranger marriages).
But, the royal families of Europe are textbook examples of the problems of recessive genes accumulating in one branch of the gene pool. They're very lucky their problems are not worse than they have been. However, the sheer number of miscarriages, stillborns and multiple child-deaths is typical of inbreeding populations (as anyone who has studied genetics must surely know).
If we were to take the recently studied 10,000 first cousins in America and have them all inbreed for the next 8-10 generations, the results of the inbreeding would be much more apparent. Statistically, it's impossible for it to be otherwise - that's how recessive genes work.
Many of us probably have some recessive genes that cause no problems in ourselves, having inherited a single gene that mutated in some great-great-grandparent. But if we marry some other descendant of that same great-great-grandparent, the chances are much higher of getting the same rare mutation.
Sources:
How to compute an inbreeding coefficient
Bittles, A.H. (2001). "A Background Background Summary of Consaguineous marriage". consang.net. Retrieved 2010, citing Bittles, A.H.; Neel, J.V. (1994). "The costs of human inbreeding and their implications for variation at the DNA level". Nature Genetics (8): 117–121.
PLoS ONE: The Role of Inbreeding in the Extinction of a European Royal Dynasty
(And many other articles).
So do not use the recent "it's not that bad to marry your first cousin" study to claim that generations of first, second and third cousin marriage does not result in a higher expression of homozygous recessive illnesses, that would be an error.