For me, it is a little from column A, and a little from column B. Sometimes the more the better, other times, not so much.
When I think about active guys with multiple title reigns, a few come to mind: Triple H, John Cena, Randy Orton, and Edge. Of these, I'd say that Triple H is not hindered by these multiple reigns, John Cena is moderately, Orton is weak in comparison, and Edge is in a ridiculous situation. I'll explain further.
Triple H: He is basically getting the Flair push. Everyone knows this. His 13 title reigns have spanned the past decade (and a little change), to the point where he averages about a title reign a year. As to the quality of these reigns, it has differed. On one hand, he had perhaps the greatest heel run in recent memory in 2002-03. On the other, he held the title for only a few hours in 2007. While this fluctuation is a bit odd, I don't think Triple H suffers because of it. If anything, he has a more menacing, bloodthirsty approach to his matches because of this constant hunt for the title.
John Cena: I don't really have a problem with Cena's reigns. They normally start, last, and end well, which has pushed him to the top. As for quality, I need not mention the year long reign back in 2006-07, nor the 2005-06 run either. While the on and off trade with Orton really boosted Cena's title numbers, he is supposed to be the face of the WWE, and the top guy deserves to have the title often. I'm against the short tradeoff title reigns when they aren't booked well, so I do think Cena suffers a tad, as he is relatively young, and is likely to be in the main event for several more years. But, it comes with the territory.
Randy Orton: Orton is a confusing case. He constantly switches roles in the WWE, going from heel to face to heel to tweener. While this isn't directly related to title reigns, it does leave me a bit confused as to where to categorize him. Either way, he is often feuding with Cena or Triple H, who both outrank him significantly in title reigns. Strictly by comparison, this makes Orton look a bit weak. Fans can do math, and how are we supposed to believe a 13 time champion will have a huge problem with a 6 time champion? I know the scripted threat is there, but sometimes it just feels like each of Orton's reigns was rushed. His first was in 2004, so he essentially averages a title reign a year. However, consider that his second reign wasn't till 2007. So, five reigns in three years. A bit much when you aren't the top guy in the business.
Edge: Edge plays a pivotal role in the WWE. He is the filler top heel, the filler top face, and the ultimate transitional champion. This is a good thing for the character, but could be a bad thing for the business, depending on your point of view. Since 2006, he has become a 9 time champion, averaging roughly 2 title reigns a year. This is about double Triple H's, and double Randy Orton's. This equates to lots of short feuds, quick title changes, and weak champions. If anything, the number of title reigns of some other guys are in direct comparison to Edge's, just because of some feuds they had together. Edge's time as champion also is incredibly short, as he is always playing hot potato with a title. It has been his game since 2006, and he isn't slowing down now.
Obviously, title reigns are supposed to be significant. Some guys get a huge boost by being a champion or former champion. Triple H is a great example of this. Flair was a great example of this. On the flip side, there are guys that have had only one title reign and have received an equally large boost. Guerrero, Benoit, and some others come to mind as guys that got elevated off one title reign, with no sequels. Every title match they were in afterward brought the feel of a potential repeat title reign. Guys like Cena, Orton, or Edge lose this feeling, as they already have won the title well over five times already. We know what it looks like when they win, and we know they can and will be at the top again.
So yes, multiple title reigns can and do degrade certain characters sometimes. I don't think it is an overall thing, more like a case by case basis. Again, a little from column A, and a little from column B.