How do I calculate the allowable manufacturing tolerance of dimensions
in parts that will be joined in an assembly of many parts, when I know
the manufacturing tolerance required of the final assembly?

Clarification of the question:
* Assume I have many bars of varying mean lengths: x1, x2, x3... xn
* Each bar is manufactured such that it is statistically capable
[http://www.isixsigma.com/dictionary/Cpk-68.htm] within a known limit
tolerance.
* My assembly consists of all bars joined end to end.
* The mean length of the assembly will be the sum of the mean lengths of the parts.
* What will be the achieved tolerance of the length dimension in the assembly?

What will be the relationship between tolerances of the parts to the
tolerance of the assembly?

My practical problem is deciding where to invest my time and effort in
reducing variation of the parts (by improving manufacturing methods)
to make sure I will have an allowably small number of rejects of the
assemblies. How do I know when the part tolerances I can achieve are good enough?

The answer should be in the form of (an) equation(s) with variables
for the means and standard deviations of the parts and the mean and
standard deviation of the assembly.
Links are good, but I have two papers on this and am unable to stay
awake long enough to get my head around them, so I really want the
equation(s).
Extra credit if I am coming at this problem from the wrong direction
and you show me how I should be approaching it (this is a dangerous
answering strategy - I have to agree with you).
Extra credit for showing me how this might be used with non-linear
dimensions (assembly might be a bearing, so parts would be a ball
between two races - variables are ball dia, race pitch circle dia,
race cross-sectional dia, and race thickness).

Assume I'm good at algebra (or know how to use the MS Excel Solver)
but bad at statistics.

This might help found it on the web.

Whenever more than one assembly is put together, new dimensions are
created as well as new distributions. Because of this, we need to look
at the addition of distributions. The distributions that exist on the
first distribution are added to the second piece and so on till the
building of the component is finished. Because the addition of
distribution is statistical in nature, we need to know some
rudimentary statistical laws so that we may use this information to
predict the best and most reasonable resolution for production.

Formulas and Calculations
There are four laws that govern the addition of distribution.

1. Law Of Additions Averages. If parts are assembled in such a way
that one dimension is added to another, the average dimension of the
entire assembly will be equal to the sum of the average dimensions of
the parts.

Let A = The Mean of Part A 
 B = The Mean of Part B 
 C = The Mean of Part C 

Average dimension of assembly = A + B + C ,etc. See Fig. 1 



2. Law Of Differences. If parts are assembled in such a way that one
dimension is subtracted from another, the average dimension of the
entire assembly will be the difference between the average dimensions
of the parts.

Let D = The Mean of Part D 
 E = The Mean of Part E 
Average dimension of assembly =( D - E )or( E - D )as the case may be. See Fig. 2 



3. Law Of Sums and Differences. If parts are assembled in such a way
that some dimensions are added to each other and some dimensions are
subtracted from another, the average dimension of the entire assembly
will be the algebraic sum of the average dimensions of the parts.

Average dimension of assembly = A + B + C - D + E ,etc 

4. Law Of The Addition and Standard Deviations or Variances. If the
parts are assembled at random, Standard Deviation, (Sigma), of the
assembly WILL NOT BE the simple sum standard deviations of the parts,
but rather, it will be the value obtained by squaring each of the
component standard deviations, totaling the squares, and then taking
the square root of the total.*

Let A = The Standard Deviation of Part A 
 B = The Standard Deviation of Part B 

Standard Deviation of the assembly =  
the square root of ((A)^2 + (B)^2 )

The forth law should be carefully examined because the statistical
addition gives a different result from the one which he/she would be
likely to get naturally.

One should note that the squares of the standard deviations are always
added regardless of whether the average dimension is gotten by sums of
differences. DO NOT attempt to subtract one standard deviation from
another as may be done in the case of averages.

The forth law can also be expressed using variances instead of sigma,
(standard deviations). Variance is the square root of standard
deviation (2). If (A)2 is the variance of Part A and  (B)2 is the
variance of Part B, the variance of the assembly will be  (A)2 + (B)2
.

Found this here http://www.sixsigmaspc.com/six-sigma-spc-articles/design-margin-statistical-tolerance-part2.html

That is gold. Thanks.
(I think there's a typo in the last paragraph. "Variance is the square
root of standard deviation" should read: "Standard deviation is the
square root of variance").