When we say "the energy in a system" (in chemistry) do we mean the energy stored in the chemical bonds in the compounds present in that system?
The answer to this question depends on the nature of the system we have defined. 

Firstly, to define the term system according to this Wikipedia article:  


  “...a physical system is a portion of the physical universe chosen for
  analysis. Everything outside the system is known as the environment.
  The environment is ignored except for its effects on the system. The
  split between system and environment is the analyst's choice,
  generally made to simplify the analysis. For example, the water in a
  lake, the water in half of a lake, or an individual molecule of water
  in the lake can each be considered a physical system.”


Every system has some relationship to it’s surroundings.  As another way of stating the above, we can describe the surroundings, or environment, as all parts of the universe that is not part of the system. We can then generally categorize the system as being isolated, closed or open:

Isolated system: Neither matter nor energy may pass into or out of the system.
Closed system: The exchange of energy is allowed but matter may not pass into or out of the system.
Open system: Both energy and matter can pass into and out of the system.  

In the context of your question (as I think I understand it), we are talking about isolated systems.  It wouldn't make much sense to discuss the energy of chemical bonds without constraining the system, even if hypothetically, in this way. To use the lake analogy, we talk about the chemical bonds within a single water molecule in a manner that excludes the matter or energy in the rest of the lake (and the rest of the universe).  If we are talking about hydrogen bonding between the water molecules for example, then we are considering there to be some hypothetical container within the lake that excludes all matter and energy from outside the container.  The contents, matter and energy, of the container, comprise the system.