Although dictionary says 'massive' only means 'heavy', I am always having the impression that in some contexts 'massive X' means 'large number of X'. Is my impression purely wrong or there are idiomatic usages like that? If there is, what is the usage? Does the phrase 'massive failures' mean 'severe failures' or 'many many failures'?
Massive literally means "having much mass" (aka very heavy). Figuratively, it is often used to mean "large."
You could never say something like "There are massive people there." (Actually, you could, but that means that there are a couple of really heavy people there; it would not mean that there are many people there.)
The phrase "massive failures" means "severe failures" as you said.
In the American English usage that I am familiar with, when you are using "massive" with a singular noun or a plural, it will always mean "large" or "heavy". For example,
Massive cat (Large or heavy cat)
Massive cats (Multiple cats, and each individual cat is large or heavy)
However, if the noun is a singular noun that refers to a group, then it can mean "a large number". For example,
Massive fleet (A fleet with a large number of ships)
Massive crowd (A crowd with a large number of people)
In a way, this can relate back to the original meaning. To make a fleet larger or heavier, you add more ships. To make a crowd larger or heavier, you add more people.