Yes, it is possible to destroy atoms, but this is not an easy task.

To destroy an atom, we need some requirements, which are not possible in an ordinary lab of physics.
There can be two possibilities to do so,

Possibility #1:

Let consider the example of the H-atom. H- atoms consist of a proton and an electron. Now to destroy the H-atom, we need two more particles, anti-proton and ant-electron. Now, if these particles are accelerated and collide with each other, then these particles disappear, and high energetic photons (electromagnetic energy) would appear instead.

To accomplish this process, we need anti-proton and anti-electron, which are not easy to obtain in a lab.

Possibility #2:

The second possibility is the removal of electrons from the shells of an atom.  The removal of electrons is only possible when the temperature increases to the millions of degrees.  At this high temperature, nuclei can be torn into fragments with high energy particles. So, at such high temperatures, an uncontrolled reaction can start.

So, both of the above possibilities seem very difficult. So, now what is the possible answer?

The answer to this question is in the law conservation of matter and the law of conservation of energy.

“Matter (for example, an atom in this case) can neither created nor destroyed” Similarly, “Energy can not be created nor destroy”

Or we can write, ” Energy can neither be created nor be destroyed, but can transform from one form to another” So, is there any relation between matter and energy? Yes! There is. Special relativity theory E=mc² that is, Energy is directly proportional to mass, which means, if we increase the energy of an object, then the mass in the object will also increase and vice versa.

What are you think; about chemical reactions? 

In any chemical reaction, new bonds form, and energy is either released or absorbed, but the total number of atoms remains the same. Again, in a chemical reaction, the total number is conserved; it is just the energy of the system, which can only change.