How do nuclear fuel rods get "spent"? Why can't you just throw neutrons at them? The idea of fuel rods being both "spent" and dangerously radioactive has always seemed to me to be self-contradictory. The typical explanation given is that the human body and environment are highly sensitive to radiation, so a below-profitable level of activity in the rods for electricity production is still enough to be plenty harmful for the people and the fishies. But why can't you throw neutrons at the rods from an external source to squeeze out more juice?
 A: The fuel in the rods is $^{235}U$, the easily fissionable isotope of Uranium.
When $^{235}U$ is truck by a neutron, it undergoes fission. In this process, the nucleus is split into 2 lighter nuclei such as strontium, neodymium and many others (some neutrons are also released, and these are used to keep the reaction going). These nuclei cannot undergo fission. Hence they are useless in providing further power to the reactor.
As a result of this, as $^{235}U$ is used up, the fuel rods can provide less and less power. In addition to that, the fission products can also absorb some of the neutrons, again reducing the power available from the rods. At some point the power available drops sufficiently that a new supply of $^{235}U$ is needed, by installing new fuel rods.
A: It is physically possible, but the technology, economics, and politics are not there yet. Maybe the economics/politics will never get there, but that's speculative.
The biggest problem is that we have plenty of uranium (and other energy sources) and burying the waste is not very expensive so there is not a large economical incentive to burn the rest of the fuel. This in turn leads to less interest in developing a reactor that can do what you suggest.
For some hypothetical examples (some of which have prototypes) see:
https://en.wikipedia.org/wiki/Traveling_wave_reactor
https://en.wikipedia.org/wiki/Breeder_reactor
You can read more about obstacles that these reactors face, including safety and economical concerns, here:
http://fissilematerials.org/blog/2010/02/history_and_status_of_fas.html
A: The nuclear fuel in a fresh fuel rod is mostly U-235, but there is a contribution from the fuel from U-238.  The uranium isotopes have very long half-lives, which means the radioactivity is relatively low.  At the beginning of life, you can stand next to a fuel rod and touch it with no issues.  The fuel rod is actually less radioactive than the uranium ore that they dig out of the ground.
Once the reaction starts, the uranium fissions and creates lighter "fission products".  The fission products have much shorter half-lives, and are therefore much more radioactive.  The longer the chain reaction goes on, the more fission products build up.  If you stand next to used fuel rod, the radiation from the fission products will kill you very quickly.  The fission products also act as absorbers, and tend to shut the reaction down.  You are slowly losing fuel and also building up absorbers.
This explains the contradiction.  As the chain reaction goes on, there is less "usable" fuel left, but the fuel rod is much more radioactive due to the fission products.  When the fuel is "spent", there is still some usable fuel remaining, but there are also a lot of fission products.
Another thing that you are not considering is that the fuel and clad undergo radiation damage, so there is a limit on how long the fuel rods can be in the reactor.
As you suggest, you could continue to throw more neutrons at the fuel to keep the chain reaction going.  This is what happens at the end of life of a fuel assembly.  The assembly does not have "reactivity" to remain critical by itself, but it gets net neutrons from other parts of the reactor to continue generating energy.   You can keep the fuel assembly in the reactor as long as it is economical and as long as you don't go over the limits from radiation damage.
Another thing you can do is to reprocess the fuel.  At the end of life, you could remove the fuel from the cladding and "split" the usable fuel from the fission products and re-use the uranium (and built-up plutonium) in new fuel rods.  The reason that we don't do this is economics.  Right now, it is much easier and cheaper to use newly mined uranium than reprocess the fuel.
