What seems fishy? The read length? While there is no official 150bp PE run on the HiSeq 2000, I know of labs that did it in 2012 or 2013. So unless you see a very strong quality drop or strange artifacts in the reads towards the end there is no reason to automatically declare them bad just because of the read length.

There's nothing particularly fishy here. You don't expect to see adapter contamination in all of your reads, at least unless you were doing small RNAseq...but then you wouldn't use 150bp PE reads to begin with. Typically only a single-digit percent of reads will show adapter contamination in normal RNAseq datasets, so it's common to have most reads remain untrimmed.