Actually anyone could register to study at either Oxford or Cambridge University. In order to be accepted at Oxford a boy/man had to pass the matriculation exams, which ensured that he had sufficient education to cope with the courses of study offered. Aparently the standard age of entry was sixteen or seventeen, and James was obviously rather precocious, as his father had been.

If that seems to suggest that neither Oxford nor Cambridge was the cutting edge of research and learning in JA's England, then it suggests correctly. The two universities were considered function manily as places where young men could connect with each other whilst preparing themselves for church, government, legal or political careers. That's not to say that there was not a high level of scholarship there- there were geniuses at both places and a lot of research was done, books were written, etc, etc. but there were other avenues outside the universities open to to investigative minds in the sciences and in the literary arts.

Just for comparison, the Scottish universities- St Andrews, Glasgow, Edinburgh and Aberdeen- took boys as early as twelve, and provided generally more career-minded training.