I apologize for not following the rules. The code for my actual problem using the hints by Bill Simpson follows. Unfortunately I seem to not be able to think of a simple example.

This is the solution to the problem I posted earlier today using the code for the actual problem. 
The functions and input paramaters are defined in another notebook.
gf\[Beta][{x_, \[Beta]_}] := 
 Module[{x1, x2, xtst, fnw, f1, xnw}, xtst = Total[x]/2; 
  fnw = LAcr[xtst, 1/12., LAI, LBI, AF, a, b, c, d, 
    400000., \[Alpha]0, \[Beta]]; 
  f1 = LAcr[x[[1]], 1/12., LAI, LBI, AF, a, b, c, d, 
    400000., \[Alpha]0, \[Beta]]; 
  Return[If[Sign[fnw] == Sign[f1], {xtst, x[[2]]}, {x[[1]], xtst}]];]
  f1 = LAcr[x[[1]], 1/12., LAI, LBI, AF, a, b, c, d, 
    400000., \[Alpha]0, \[Beta]]; 
  Return[If[Sign[fnw] == Sign[f1], {xtst, x[[2]]}, {x[[1]], xtst}]];]`
f[x_] := gf\[Beta][{x, .6}]
t works slightly differently from previously.  What used to work, doesn't.
In[27]:= NestWhile[f, {500., 50000.}, test &]

Out[27]= {500., 50000.}
However, if I add a large integer it does.
In[28]:= NestWhile[f, {500., 50000.}, test &, 100]

Out[28]= {35964.7, 35964.7}
The integer seems to be the number of iterations.`enter code here`
In[29]:= NestWhile[f, {500., 50000.}, test &, 3]

Out[29]= {25250., 37625.}

Usually the & is only used to indicate the end of an "anonymous" function that hasn't been previously defined. You can look up "Function" in the help pages and see what you can make of that. It is sometimes confusing for new users because they are several different notations used to accomplish the same thing and this can take a while to figure out.

It is also the case that it is sometimes difficult to figure out exactly what some fragment of Mathematica code might do, even if you are using it in a fairly conventional and expected way. That difficulty is magnified substantially if you start using something in an unconventional and unexpected way.

Mathematica provides the ability to Trace the execution of code using Trace[...your Mathematica expression...], but it may take a little experimenting to understand the output from Trace. If you can figure that out, perhaps starting with very simple examples and working your way up, then this might help you understand what is happening and what is going wrong in your actual code.

The fourth argument to NestWhile is described in the help pages, particularly if you click on the "Details" or sometimes labeled "Details and Options" near the top of the help page. When you think you understand NestWhile then you might try creating some very simple examples and explore whether it acts as you expect.

Beyond this I am at a disadvantage because I don't have a description of what you are really trying to accomplish and don't have a complete set of code to experiment with to check whether my suggestions even make sense.

I'm glad that what I wrote earlier gave you some assistance.