not even 300 bp.

Not surprising. Libraries can have inserts shorter than the length of sequencing. There is a distribution of insert sizes in any given library. Once sequencing runs out of real sequence, sequencer goes on to read into the the adapter at 3'end. If experimental people expected these inserts to be longer then they seem to have missed the mark.

I also observed that those reads with the shorterst length have more abundance

Shorter inserts always tend to cluster better. Probably partly due to physics since the initial binding/clustering is a random process. That is the reason one needs to get rid of adapter dimers etc since they will out compete real library fragments.

Can one trust these runs ?

Why not? Trim the data, stat analyzing the reads. We don't know what this experiment is about but as long as there was no other issue (hardware/software/kit) with the sequencing the data should be fine.