To add to Benito Caro's amusing, evocative and quite accurate analogy.
The notion you cite that atoms can't be cut died about 100 years ago. There are two reasons this idea sometimes persists in modern culture and, although wrong, used the right way, it can be useful:
The classical Greek name "atom" literally means "not cut" and thus refers either to something unhewn (i.e. uncarved, unprocessed wood or stone) or to something that cannot be cut. ("a-" being a negative prefix, and "temnein" to cut or hew, "tomos" present participle "a cutting": c.f. also "tomography": literally "a drawing of a cut [view]"). So to call anything that can be broken into a smaller bits an "atom" is etymologically if not from a strict logical standpoint an oxymoron. The ancient Greek philosophers argued theologically (rather than from experimental evidence) that Nature comprised indivisible parts they called "atoms" and the idea took deep root in Western culture. It's been around a long time. You can find it in the writings Chaucer and Shakespeare, just to name a couple of linguistically "parochial" examples I can think of. So, it was always on the cards that the first scientific usage of "atom" was a risky business likely to be shown to be too keenly applied when knowledge advanced, which brings me to the second reason for the idea's persistence:
Although "atoms" can indeed be cut, they are hard to cut; either that, or everyday materials do not have atoms that spontaneously break apart (i.e. are radioactive). Therefore, people did not notice their cutting until after the name had been applied by scientists, notably John Dalton in the early 1800s. Scientists recognized that elements reacted such that the amounts consumed by reaction were always rational ratios of small "unit" numbers, hence his hunch that chemical reactions involves the rearrangement of "indivisible" things whilst leaving the "indivisibles" intact. The notion of atoms being indivisible defines the science of chemistry: most of the interactions of the everyday world are interactions wherein the atoms stay in one piece and the electrons around them re-arrange themselves. That is, chemical reactions happen, and materials change their states. The energies needed to begin reactions where the atoms themselves break apart are very, very high by everyday standards, hence these reactions are highly improbable in everyday observations, hence they were not observed until the early 1900s. I have a chemistry textbook from 1904 which clearly states on its first page that "atoms are in-principle indivisible"and so my book represents roughly the end of life of the "a-tomos" notion applied by John Dalton one hundred years earlier.
So the notion of "a-tomos" is a useful one when doing chemistry: the atoms are indeed atoms of chemistry.
Nowadays we seem careful not to make the same mistake, so the name "atom" has been left where it is and we call the atoms of our modern theories "fundamental particles" rather than "atoms". These are quantum states that live in invariant, indivisible subspaces of a larger quantum state space in all of the interactions we observe. They are, to our knowledge, the quarks, the leptons and a handful of others.
Energy changes arising from the rearrangements of bits of atoms are huge compared with those arising from rearrangements of atoms themselves that leave the atoms intact. This observation is also linked to why the speed of light seems so high to us: the speed of light defines (through Einstein's famous rest mass formula) the order of magnitude of the ratio of speeds and energies of nuclear reactions to those of everyday processes like chemical reactions and heartbeats that give rise to our "slow" units of measuring time.
So why atoms are so hard to cut is roughly, ultimately, the same reason that light seems to move so fast.