What can they do? It's not Twitter themselves being DDOS'd, it's a DNS provider. This propagates up the chain to impact both a Tier 1 network and cloud providers, which hits tons of stuff on top of that.
host -t ns twitter.com: ns3.p34.dynect.net, ns4.p34.dynect.net, ns1.p34.dynect.net, ns2.p34.dynect.net.
host -t ns amazon.com: ns3.p31.dynect.net, ns4.p31.dynect.net, ns2.p31.dynect.net, pdns6.ultradns.co.uk, pdns1.ultradns.net, ns1.p31.dynect.net.
"How do all these major players have singly-homed DNS"?