Can any enzyme be produced?
No, not all enzymes (or other proteins for that matter) can be obtained in functional form by recombinant expression with today's methods. As you suspect, problems arise when complex post-translational modifications are necessary to obtain the correct function.

Direct modifications to peptide is one potential problem. Some of these can be resolved by expression the recombinant protein in the right cell type: for example, expression in a bacteria may not provide the right modification, but expression in yeast or a mammalian cell might. CHO cells are a common mammalian cell type used in such cases; antibody production in hybridomas is another well-known example. But there are plenty of posttranslational modifications that are poorly understood, and most likely there are still many completely unknown types; new ones are still being discovered.

Another posttranslational event is association with cell membranes. Membrane proteins (including membrane-embedded enzymes, for example in lipid metabolism) can be very difficult to express properly, as not only the peptide but some surround membrane structure must be reproduced in the expression system to obtain correct function. This is also a major hurdle in obtaining structures for membrane proteins, notably many cell surface receptors. This article is a summary of the current challenges and unsolved problems in this area. And you can imagine the problems with proteins that need to form large superstructures to be functional --- large protein polymers, respiratory chain complexes, ribosomes, DNA replication forks ... 

So there is no universal method; all proteins are different, some are easy to express, some are extremely hard.