Personally, the 'starts off nicely and then loses its luster' types of stories are really annoying. They'll have characters that tu like, an en general, general nice plot, with genres of your preference... but nothing exciting will take place! Or, even if it does, it'll usually be something insanely boring/predictable/pathetic.
This has happened to me many times. Especially in FanFiction, since I don't read libros that much. I get attracted to the summary and the pairing, then I become hooked with the early chapters... then a horrible twist comes along and literally ruins everything the autor has worked on so far. The story's interesting-ness (is this even a word?!) pretty much goes downhill after that. Unless the autor manages to handle the situation cleverly enough. Which is rare.
The saddest thing about these kinds of stories is that they have lots of hidden potential within them, but the author's not executing the scenes well enough. How depressing.
The 'boring at first, but becomes exciting later' stories aren't that bad for me. I usually skip the boring chapters and read through the parts where the fun starts. Then, if the anterior chapters hold important background information, I'll read them just for the sake of understanding the characters better.
Sometimes, they're not as boring as they appear. It could be the overload of difficult words that make us feel intimidated and uncomfortable.
The excitement later on usually makes up for the earlier boring junk. At least, in my opinion.