I have a PhD in Applied Math from one of the Math departments in the country, and now do biomedical research. Believe me, the cynicism is well placed.
I wanted to get a PhD because I believed as you do. The fact is, though biomedical research is a field where cheating can take you a long, long way. The experiments have gotten so big and expensive that scientific accountability is at an all-time low (for example, check out this commentary [1] on pre-clinical cancer research. And the money in biomedical research attracts power-hungry people. At the same time, the pressure to publish in order to obtain grant funding is enormous, and the incentives to cheat to get ahead by e.g. playing games with credit or retarding competitors with sly reviews is correspondingly intense.
[1] http://www.nature.com/nature/journal/v483/n7391/full/483531a...
TLDR: drug company attempts to replicate results of high-impact cancer
biology research papers, in a search for new drug targets. Failed to
replicate the claims in 47 out of 53 papers. Authors refused to
provide materials necessary for replication until company signed NDA
stating they would not publish the results of the replication
attempts. One author tells the head of the replication effort "Well,
we only got this result 1 time out of 5 repetitions, but we decided to
report the 1 instead of the other 4, because it's the better story."
(NDA story is from the subsequent Nature clarification, sentence 4 is
from a news report on the commentary.)