Are 1 bit full adders made up of 2 half adders, and then the say 4 bit adders made up of 4 1-bit full adders?

Am I getting this correctly?

 

Yes, four bit adder is 4 one bit full adders with connected Cout to Cin of the next stage.
They say? Shouldn´t this be obvious?

 

Are 1 bit full adders made up of 2 half adders, and then the say 4 bit adders made up of 4 1-bit full adders?

Am I getting this correctly?

Not quite. Half-adders are not combined to make full adders. A half-adder has two inputs ( A, B ) and two outputs (S, CO). It is commonly used for the first stage in a chain of adders where there is no CARRY IN. A full-adder has three inputs (A, B, CI) and two outputs (S, CO). The full-adder is used for all the stages except the first stage because there is always the possibility of a carry in for stages 1..n, but there is no carry-in for stage 0.

http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/Comb/adder.html

 

Not quite. Half-adders are not combined to make full adders.

Not quite.
See for example here: http://www.tpub.com/content/neets/14185/css/14185_125.htm

A full adder constitutes of two half-adders and one OR gate to be exact.

Also might be useful http://www.play-hookey.com/digital/adder.html and http://en.wikipedia.org/wiki/Adder_(electronics)