Possibly, but not on the free tier. Anyway, our current plan can only handle 20 concurrent connections to Postgres and I am not planning to pay more to satisfy spikes.

Heroku doesn't automatically scale it for you, though you have the ability to increase the number of dynos. There are services that will do that for you, but it's probably better that you have to explicitly scale so you don't get hit with a sizable bill you weren't expecting.

In any case, there are other constraints than just the amount of webservers that could be causing the outage e.g. the database.