Are there uncountably many subintervals in $[0,1]$? Are there an uncountably infinite number of sub-intervals in $[0,1]$ such that the number of real numbers in each of those sub-intervals is uncountably infinite?
I would say no, because you would have to make the intervals really small thereby making it impossible to have an uncountably infinite number of real numbers in those sub-intervals, but I'm no expert.
 A: If the intervals have to be disjoint then since each contains a rational number you can map the set of intervals injectively to the rationals.  So the set must be countable.  Injectivity follows from the disjointness.  
If the invervals are allowed to overlap then $(a,1]$ is a subinterval for each $a\in[0,1)$ so is an uncountable set.
A: Just note that $[x,1]$ is an uncountable interval for all $x<1$, and there are uncountably many $x\in[0,1]$ such that $x\neq 1$.
It is true that there are no uncountably many pairwise disjoint intervals, because from each interval we could pick a different rational number, and therefore create an injection from that family of pairwise disjoint intervals into a countable set, so such collection must be countable.
A: There is an uncountable number of intervals of the form $[0,\alpha]$ where $\alpha$ is an element in $(0,1]$, each of these contains an uncountable number of elements.
If you want them to be disjoint then it is impossible, because every non-empty interval will contain a rational number. And there is only a countable number of these.
A: The question doesn't make sense as written, do you want the intervals to be disjoint? It is false if you want them to be disjoint because every interval on the real line must contain a rational, so therefore you can only have a countable number of intervals (as each one contains a different rational number). 
If you just want like disjoint sets, it is clearly true because $[0,1] \cong \mathbb{R}\times \mathbb{R}$, so you can set each class ${a}\times \mathbb{R}$ as an uncountable set in $[0,1]$ and they will all be disjoint. This is why making them "really small" doesn't work in general, but does work for intervals!
