I would say trust your own gut. For example, last year I only applied to a handful of law firms (like 5), as I was busy with uni. This year, I am on a gap year, so I have plenty of time to make as many high-quality applications as possible.
Personally I think it's very possible to do lots of written applications and have some success. The real difficulty comes after when trying to prepare for any type of interview. Trying to properly prepare for, and tailor interview answers to multiple firms is really difficult. If you've got 20 applications on the go you're simply not going to be a stand out candidate for them all. Whereas if you'd applied to 5 firms, you could really spend time knowing those firms inside and out, giving really personal and tailored answers to any question. As with anything, it really depends.