Combining then sorting Javascript arrays. Getting unpredictable results

That's because you're not stripping off the surrounding spaces, especially the leading spaces. The sorting is off because a space comes before any letter. The below code should fix it, assuming the whole line has no surrounding white space:

// split on pipe and surrounding white space
var splitRe = /\s*\|\s*/;

carriersOne = carriersOne[0].split(splitRe);
carriersTwo = carriersTwo[0].split(splitRe);

You need to trim the spaces:

carriersOne = carriersOne[0].split('|').map(function(e){return e.replace(/^\s*/, '')});
carriersTwo = carriersTwo[0].split('|').map(function(e){return e.replace(/^\s*/, '')});

The problem is that you have leading spaces in the carrier names.

     ["    Some Other Company", " Chicago Insurance Company ", " Hospital Insurance    Corporation", " New York Health Care Insurance Company ", "Advantage Care ", "St. Joseph's Medical Center "]

You can remove the leading spaces and get the result you're looking for like this:

carriersOne = ['St. Joseph\'s Medical Center | New York Health Care Insurance Company |    Some Other Company'];
carriersTwo = ['Advantage Care | Chicago Insurance Company | Hospital Insurance    Corporation'];

carriersOne = carriersOne[0].split('|');
carriersTwo = carriersTwo[0].split('|');

allCarriers = carriersOne.concat(carriersTwo);
allCarriers = allCarriers.map(
                 function(carrier){ 
                       // replace leading and trailing spaces
                       return carrier.replace(/^\s+|\s+$/g,'');
                 }
               );
allCarriers.sort();
console.log(allCarriers);

It's the spaces in between the pipes. Here's a much fancier (and more flexible) solution.

http://jsfiddle.net/THEtheChad/3v7ss/

var carriersOne = ['St. Joseph\'s Medical Center | New York Health Care Insurance Company |    Some Other Company'];
var carriersTwo = ['Advantage Care | Chicago Insurance Company | Hospital Insurance    Corporation'];

var allCarriers = parseAndMerge(carriersOne, carriersTwo).sort();

var count = allCarriers.length;

allCarriers.forEach(function(carrier){
    alert(carrier);
});

function parseAndMerge(){
    var args = Array.prototype.slice.call(arguments)
      , collector = []
    ;//var

    args.forEach(function(arr){
        collector = collector.concat(arr[0].split('|').map(trim));
    });

    return collector;
}

function trim(string) { return string.trim() }

The spaces in between the pipes | are causing your sort order to be incorrect. Try something like this:

http://jsfiddle.net/THEtheChad/LQxW8/

carriersOne = ['St. Joseph\'s Medical Center | New York Health Care Insurance Company |    Some Other Company'];
carriersTwo = ['Advantage Care | Chicago Insurance Company | Hospital Insurance    Corporation'];

carriersOne = normalize(carriersOne);
carriersTwo = normalize(carriersTwo);

allCarriers = carriersOne.concat(carriersTwo);
allCarriers.sort();

count = allCarriers.length;

for(i=0;i<count;i++) {
   alert(allCarriers[i]);
}

function normalize(arr){
  var carriers = arr[0].split('|')
    , i = carriers.length
  ;//var

    while(i--){
      carriers[i] = carriers[i].trim();  
    }

   return carriers;
}