Why do adapter sequences sometimes appear somewhere other than 3' or 5' end?

[jimmybee]

Why not just attach a longer adapter for the shorter reads so this error does not happen?

Edit: Or just use some other adapter that is made up of some other chemical so it is distinguishable from the insert?

This isn't an error, its a limitation. A lot of applications of genome sequencing calls for sequencing a fragment shorter than the read-length (e.g. aDNA, small-RNAs). Adapter trimming corrects for this. It doesn't matter if you have a short to long adapter, you'll still get this issue.

You can reduce the adapter contamination by doing a more stringent fragment size selection before sequencing...

[prs321]

Why not just attach a longer adapter for the shorter reads so this error does not happen?

Edit: Or just use some other adapter that is made up of some other chemical so it is distinguishable from the insert?

[mastal]

the fragments are not all exactly the same length, while most
may be close to the average size, some are longer and some are
shorter.

for fragments that are very short, you read into the adapter
somewhere in the middle of the read because you have read through
all of the insert.

there are also the occasional reads made up of adapter polymers
only, so they have the adapter at the 5' end too.

[prs321]

How does this make any sense?

If I take the DNA of some organism and get it sequenced, I first have to chop it up into many fragments.

Then these fragments have adapters attached to them.

How does it make any sense that you can have an adapter in the middle of a sequence if you attached an adapter on both ends in the first place before sequencing?