There is a range of expectations that more or less lines up with the expectations and needs of the faculty at the institution. If half the faculty require programming skills to complete a PhD under their guidance, then the lack of programming skills will be a barrier.
Also, there is an evolving understanding of what "programming" means. Some mean more traditional languages like Fortran and C. Others include skilled use of higher level tools like MMa, MATLAB, etc.
The fewer tools a student has in his box, the narrower their options.
I can't really recommend any scientist, engineer, or self-respecting math geek try and blaze an educational or career path without some programming skills in their tool box. Your options will be too narrow and only get narrower over time.