Commercial database systems versus open source database systems
This really depends on your requirements. Many systems built on top of open source products are rock solid.

Note that your example does not impress me - Twitter is not reliable at all by my customers' standards - it has frequent downtimes, it loses tweets and followers, its search is not 100% correct etc. I work in finances, and my job is to develop robust and performant systems. In some (but not all) of the scenarios I encounter, open source PostgreSQL could get the job done just as well as closed source Oracle, in terms of speed and robustness.

If your needs are more or less standard, than you can save a lot of precious time if you go for Oracle or SQL Server, as you might spend less time on setting things up and maintenance. As such, your investment in licensing costs may pay off quite soon.

If, however, your needs are not completely mainstream, then open source is your friend. Open source products can innovate much faster. You don't have to persuade someone that your scenario is not "convoluted", you can get a fix right away instead of waiting for years. If you really need something, you can do it yourself.