I can think of two reasons why we need a lower bound, one statistical, one inuititive.
First, the intuitive: The annihiliation/creation operators represent adding/removing particles (or excitations, or whatever). The vacuum state as lowest energy state represents obviously the empty state from which no further particles can be removed. It is also clear (in the case of bosons) that I can just keep adding particles and nothing stops me from doing that, so an upper bound of energy would mean that there is a state $|\text{max}\rangle$ on which the creation operator acts as $a^\dagger|\text{max}\rangle = 0$. This is, from an intuitive point of view, obviously nonsense. Why should adding a particle destroy all the others? Where has all the energy gone? Demanding a general upper bound for energies is thus a bad idea. [I want to stress again that this is only heuristic. It is not meant to translate to a rigourous argument directly.]
Now, the statistical. We know that the time evolution drives systems towards the state with the lowest (free) energy. If the Hamiltonian is not bounded below, there is no global minimum of the energy, and thus there are only metastable states. But since there is no ground state, such a system would be able to drop to ever lower energy levels. In consequence, a system not bounded below would be able to radiate energy infinitely. That is obviously nonsense, no physical system can hold an infinite amount of accessible energy. So though the physics can describe such a system, its time evolution behaviour is not something we observe in the real world.
Both arguments do not prohibit that there are systems with upper bounds. The easiest example are pure spin systems: Take a number of particles with spin an put them in a (constant) magnetic field. Now the lowest energy state is when all spins are flipped one way, and the highest energy state is when all spins are flipped the other way. You cannot add further energy to this system, it is naturally bounded below and above.
