Glass fibers can be bent when a sheet of glass would break. Is the strain less in a fiber? Is it deforming plastically or elastically? Is this the reason metal wires don’t work harden and break?
Try this:
Get a pencil and a can of soup
Place the flat end of the pencil on a table (so the pencil is standing upright). Tilt the pencil over by, say, 30 degrees. Measure the gap that has opened up between the table and the pencil end - it’s a few millimetres
Stand the soup can on the table and tilt it by 30 degrees, and the gap that opens up is several centimetres.
If instead of tilting objects, you were bending two pieces of some material by 30 degrees, they would both have to stretch on the outside radius of the bend (and compress or pucker on the inside, but we’ll ignore that for the moment). The thin piece of material would have to stretch by millimetres on the outside, the thick piece would have to stretch by a considerably greater amount - perhaps breaking in the process.
To expand on Mangetout’s answer, the reason fibers allow that sort of thing to happen is because they can slide past each other.
In a solid object under bending, the material on the outside and inside of the bend is forced to strain by its attachment to the material next to it. The ability for local shear forces to transmit across the width of the object ensures that the material on the outside of the bend will have to take up the entire increase in length.
In a stranded object, however, each strand can move independently and isn’t pushed into straining by its neighbors since they can slide past each other. So in the case of a simple one-direction bend in a piece of stranded cable, the net result is that the ends of all the strands don’t have to stay in the same plane. If the cut end started flat, with all the ends flush, the bent piece will have strands the poke out farther than others. These will be the strands that run through the shorter inner radius of the bend.
Cohesive forces in a solid object would keep all those “strands” glued together and forced to stay together. If they have to stay matched up over their whole lengths, they do so under bending by stretching or compressing to adjust to the newly required path length. And that kind of strain typically means breaking for brittle materials.