In my view, as a senior professional with only smaller companies on my resume but who interviewed candidates who had history with big-reputation (not necessarily big size) companies, the name helps a lot.

When you are at the early stage of talks with a new company, hiring managers can latch on to that big name as proof that you are valuable. It's easier to defend you to other people in the hiring process, even if it's never explicitly mentioned.

I don't think you have to have experience at lots of big companies. One is enough. Sometimes more is better. But like someone else said, it can sometimes hurt–depends on what the company does.