  Associated with the cosmic acceleration are the old and new cosmological
constant problems, recently put into the more general context of the dark
energy problem. In broad terms, the old problem is related to an unexpected
order of magnitude of this component while the new problem is related to this
magnitude being of the same order of the matter energy density during the
present epoch of cosmic evolution. Current plans to measure the equation of
state certainly constitute an important approach; however, as we discuss, this
approach is faced with serious feasibility challenges and is limited in the
type of conclusive answers it could provide. Therefore, is it really too early
to seek actively for new tests and approaches to these problems? In view of the
difficulty of this endeavor, we argue here that a good place to start is by
questioning some of the assumptions underlying the formulation of these
problems and finding new ways to put this questioning to the test. First, we
calculate how much fine tuning the cosmic coincidence problem represents. Next,
we discuss the potential of some cosmological probes such as gravitational
lensing to identify novel tests to probe dark energy questions and assumptions.
Then, motivated by some theorems in General Relativity, we discuss if the full
identification of the cosmological constant with vacuum energy is
unquestionable. Also, we point out the relevance of experiments at the
interface of astrophysics and quantum field theory, such as the Casimir effect
in gravitational and cosmological contexts. We conclude that challenging some
of the assumptions underlying the cosmological constant problems and putting
them to the test may prove useful and necessary to make progress on these
questions. (Abridged)
