We have over 5000 language on Earth as of now, some extant and others not. These all came from what we now call proto languages, but do scientists believe that all proto languages came from one "mono-language" or several different language origins. Ergo are there multiple "language trees" so to speak?
The question raises three terminological issues: what is "language", what is "invented" and what is "once"? It does presuppose that there was a prior state without language, and a later state with it (no controversy about that). To answer the question, we need to understand exactly what is being asked.
Language is one aspect of a more general set of cognitive tools unique to humans. Some parts are clearly learned, and others are clearly genetically inherited (those who deny a language-specific genetic component nevertheless do not deny that human cognitive abilities are genetically transmitted). "Language" is a myriad of evolutionary developments such as the physical enhancements which allow us to make speech articulations; the ability to retain an open-ended and replicable set of articulations (we don't just learn 12 calls, and we can easily reproduce the hundred and thousands of calls that we make – not by memorizing acoustic patterns of whole words); the ability to combine sequences of units into higher-level units; the ability to do this recursively; the ability to treat order of elements as distinctive... and many facts that are plainly about language, such as "grammatical agreement", "movement", "palatalization".
One can always conjecture that there was a single evolutionary event, via a single linguistic Eve, which catapulted mankind from chimpanzee-like linguistic abilities to having full grammars, but that is more a sci-fi movie storyline. The various genetic abilities that resulted in our current or pre-current state must have happened over the course of many generations. And it is highly improbable that humans had these latent abilities – to create phonemes, words, sentences, and unbounded dependencies – without actually ever doing this, until one day somebody got the idea "I should invent language". Forming and fleshing out the idea "I should invent language" cannot be done without language, in some form.
It's not clear what "once" would mean, in an evolutionary sense. If we take a hereditary genetic mutation to be the causal spark for the development of a new trait, "once" can be limited to a part of a human lifespan: there may be a single mutation that brings about a change, transmitted to a being's offspring. As I said, it is extremely unlikely that there was just one mutation that created the full underpinnings of language. However, you could look at "once" from a grander perspective, summing up over hundreds of thousands of years of human development. You can, effectively, answer the question in the affirmative simply by adopting a broad definition of when "once" is, as opposed to "twice".
Since language involves both genetic and learned facts, and in light of the fact that the question is about language being invented (not developed), we could narrow the question by focusing on the learned aspects, presupposing the prior existence of requisite genetic abilities. An example of a learned aspect of language is the set of sounds used to communicate something. Somehow, particular sound sequences got associated with real-world situations like "water; food; bear; tiger; rabbit; sex". The learned ability to create words (stable correlations of signal and referent) must have preceded the learned ability to create sentences out of multiple words, which must have preceded the learned ability to distinguish event by ordering words in a different fashion. I would say that it is inescapable that different learned aspects of language developed at different times (thus, "more than once").
There is a related question about "single common language", which can't be answered. We can reasonably suppose that in the way-olden days, there were many groups of humans with developed languages. It is then possible that at some point, all of those languages were obliterated (perhaps all their speakers died off), and thus every human language is the result of linguistic change from that single surviving language. Thus every existing language could be, accidentally, descended from one language.
There is a theoretical experiment (which will never get IRB approval), where infants are completely and absolutely deprived of all forms of language, with the question being whether the infants could develop full-blown languages. Poto and Cabengo-like situations differ from this imaginary experiment because in the former case, the stimulus is impoverished, but not totally lacking. Even performing the experiment would not tell you if language was invented twice, but it would be evidence that it could have been.
Well, it happens even nowadays that languages are created anew from fragments of other languages.
There are creoles like Papiamento or Tok Pisin that are created from fragments of different and unrelated languages and have no clear antecessor. There are also constructed languages like Loglan or Lojban that did not evolve from any natural language.
I also remember (but I could not retrieve a quotable reference in short time) a story about twins raised in circumstances of language deprivation who developed their own language allegedly not related to the language of the country they lived in.
Having said all this, the main reason for having unrelated language families is time depth: Any traces of relatedness are completely wiped out by linguistic changes over time.