Computation is effectively computable in theory and in practice
The only definitions of the term "effectively computable" that I can find online are that the phrase just means "computable" (which is usually taken to mean "Turing-computable"). As such, computability is an absolute concept: either a function is computable or it is not. Are you asking about whether a function that is computable can actually be computed in practice? If so, the answer is clearly no: it's easy to construct families of problem instances that would require more time than the expected lifetime of the sun, for example. For example, take large instances of EXP-complete problems: $2^{100}$ exceeds the age of the universe in nanoseconds.  (That is, a 1GHz computer that had been running since the beginning of time wouldn't have executed $2^{100}$ instructions yet.)