New characters can in principle be coined at any time, just as new words can be, but they may not be adopted. Significant historically recent coinages date to scientific terms of the 19th century. Specifically, Chinese coined new characters for chemical elements – see chemical elements in East Asian languages – which continue to be used and taught in schools in China and Taiwan. In Japan, in the Meiji era (specifically, late 19th century), new characters were coined for some (but not all) SI units, such as 粁 (米 "meter" + 千 "thousand, kilo-") for kilometer. These kokuji (Japanese-coinages) have found use in China as well – see Chinese characters for SI units for details.
Is there an answer to this question (If it cannot be answered, say "unanswerable"): What can be coined at any time?
New characters