You are misrepresenting the concept of undecidable. If the compiler can say if the program compiles or not, then it is most certainly decidable. What you want to say is that it cannot be determined without full parsing, so no preprocessing is possible.
No, it's actually undecidable. C++ templated have been determined to be turing complete, which means that template instantiations can encode the halting problem. Determining whether a program compiles or not therefore requires solving the halting problem.
In practice, compilers work around this by limiting template instantiation depth.
I gave an example of a template program to show the general method. Obviously, primality is decidable, but there exist candidate C++ programs whose parse tree is undecidable. The trick would be to encode your parser in a template, run it on the undecidable program (i.e., itself), and create a contrary result. Does this have any effect on practical C++ builds? I honestly have no idea.
