First we need to understand what exactly you're asking. Every time someone sends a packet on the Internet it consumes some of the potential bandwidth of the Internet. Just like when a car drives down a road, no other car can drive in that same spot, so you have one less available spot on the road.
If we continue that comparison, a single site sending out many cars can easily jam the roads around it, watch the madness after any sports game is over and all the fans are leaving at the same time. But this only causes very localized disruptions, the same is generally true of the Internet. But the attacker used an amplification technique to do more congestion damage than they could have done alone. This employed unwilling 3rd parties, who had specific vulnerabilities. This allowed the attacker to generate much more congestion than they could have alone.
So there are two more important questions, which you will never find in the news as they're technical. The news is almost exclusively concerned with dire predictions that average people can understand, and the more sensational the better, regardless of accuracy.
- Could an attack be constructed with such scale as to affect a large portion of Internet traffic in a noticeably detrimental way?
- Are there vulnerabilities in key systems of the Internet which can allow relatively little attacker effort to result in the average users' experience being disrupted meaningfully.?
As to the first question, it's highly unlikely. As you've seen attacks can be constructed which will disrupt certain very localized portions of the Internet. Spamhaus represents a very small portion of the overall Internet, and the attack was mitigated largely by a single other Internet entity. The attacker had significant resources compared to most potential attackers and was still not able to fully overcome Spamhaus and Cloudfare collaborating. As fun as it might be to theorize what might be possible, it's more than one giant leap from taking out Spamhaus to afflicting the Internet as a whole.
To the second question, this is somewhat more plausible. The Internet relies on a variety of key infrastructure. Many times security takes a back seat to functionality, especially where there are strong economic forces (ie, cheaping out). A great first example is BGP, which is what helps connect the many individual networks that make up the Internet. BGP was accidentally "hacked" by a Syrian ISP in 2012, which ended up disturbing a large portion of IP Routing. This wasn't malicious, was quickly fixed (quickly considering the scope), but illustrates how a small corruption can cause massive problems if systems aren't secured against such.
If I could predict the next such occurrence, I would be a millionaire. The best defense we have now is thoroughly analyzing systems for vulnerabilities, understanding them, developing best practices to guard against them, and vigilantly implementing appropriate measures.