Any time you recycle anything there's waste in the process. Assuming the entire jug is recyclable, you'll loose a certain percent of the mass of the plastic by melting or grinding it to be used. Even if you cut it you still lose some of it.

Polymers (plastics) become strong through a process called cross-linking.

A polymer is a ultra-long molecule. Think of a regular molecule as a pea, where as a polymer is a strand of noodles.

When a plastic is poured into shape, cross linking ties every noodle together where they touch. This gives the plastic a fixed shape.

When you recycle the plastic, either through heating them (for thermo-plastics) or melting them with solvents like acetone, most of the polymers are conserved. That is the macro-molecules remain long.

However the properties that allow cross-linking between the polymer strands is severely degraded.

I will give you an example. DVD cases made out of recycled plastic are brittle and break easily. They are utterly shitty for the job of keeping your DVDs safe.

Some plastics are more recyclable than others. Some plastics are strong enough that even at 50% strength they are still useful. MOST are not.

For example, recycling plastic milk jugs would see milk jugs shredded and then combined with 50% new plastic to produce a milk jug that is 50% recycled plastic.

I've found that lids and caps are pretty bad since they're often made from #5 plastic over #1 or #2.