A simple pyramid scheme, where the person at the top gets their pay out from the people at the bottom (the new participants), has two important characteristics:
- The number of new participants to be recruited by a participant
- The number of levels between the top and the bottom
For instance, take a look at this scheme, where the number of new participants to be recruited is 6:

Let's say you get in pretty high up, at the third level. Also, let's assume it only has three levels, so you pay the person on level one, who is then retired from the pyramid. The person above you, who recruited you, gets paid out by the people you recruit, and you get paid out by the recruits of your recruits.
As far as pyramid schemes go, that's fantastic. Usually the number of recruits and the number of levels between top and bottom (between paying to get in and getting paid out yourself) are both larger.
Also, you got in at the beginning, at level 3 overall.
But — assuming that the pyramid grows evenly — even here, there need to be over 7,500 participants in the scheme, before you get your pay out. And you got in early! Had you gotten in only one level later, or would the number of levels between top and bottom be one more, as you can see, the number of total participants needed for you to get your pay out increases drastically.
TL;DR:
(...) they brag about income being "unlimited" in theory since you could be on the bottom and still bring in enough people to make a satisfactory income (...)
No. This is not true. The pool of people who are willing to participate will be exhausted pretty quick.