Many synthetic and natural fibres can be stretched, as we can see in the making of polymer fibres:

The main difference between rubber and other fibres is that rubber goes back to its previous shape and size. We can see that stretching a fibre aligns the polymer chains. In fibres, this alignment allows forces such as those of hydrogen bonding between chains to have an increased effect, and they will be strong enough to hold the fibres in their stretched, aligned position. In rubbery polymers (elastomers), we find there are large and bulky groups along the chains, and these prevent the chains from packing together so closely. As the chains are further apart, there are not the same forces between them to keep them in the uncoiled arrangement. This weak interaction between polymer molecules is not enough to keep the rubber in its stretched position so it reverts to the original coiled state.

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