If both of the internet connections are actively used, DNS Load balancing for example, they will most likely both get crashed. Chances are smaller because of the higher bandwith, but considered todays ddos attacks being 60 to 300+ Gbit/s traffic it wont change anything in nearly all cases.
If the second connection is only used as a kinda "spare" connection as u said, the core problem still remains: Your systems crashes or drops everything. After all someone has to react and check/repair either way.
There are also proxy solutions with high bandwith availabillity, where the traffic is checked and then forwarded. The advantage is your infrastructure wont get hurt, and small attacks can be withstood. But the same problem as above stays, if the proxy gets crashed due to the traffic ur still ****** and have to react. Or in better words the company you pay for the proxy has to react.
after all, there will never be a 100% effective way to deal with it.
the dreams of yesterday are the hopes of today and the reality of tomorrow.
-Robert H. Goddard