Q:

Would this number make a good 2048 combo?

Inspired by xkcd.
Your challenge is to determine whether a number would make a good combination in the game 2048. Your input will be a number, such as:
8224

And the output will be whether that number would make a good 2048 combo, which for this input would be true or yes or 1 or any other way of indicating a positive result.
For those not familiar with the game, here's a simple explanation: powers of two are arranged on a grid, like this: [2] [2]. Tiles can be moved in any direction, and if two identical tiles meet, they become the next power of two (so [2] [2] when moved left or right becomes [4]). Or, you could just try the game here.
What does "a good 2048 combination" mean? It means any number that, if it was in the game "2048", it could be combined into one single number. (A zero means an empty space, and can be ignored if needed.) Note that numbers could possibly be multiple digits! However, the numbers must not change between moves. Here are some examples/test cases (with "Good" indicating a good combination, and "Bad" meaning not good):

Good: 8224 (8224 -> 844 -> 88 -> 16)
Good: 2222 (2222 -> 44 -> 8)
Good: 22048 (22048 -> 448 -> 88 -> 16)
Bad: 20482 (cannot combine the outer 2's, nor can you combine a 2048 and a 2)
Good: 20482048 (20482048 -> 4096)
Bad: 210241024 (210241024 -> 22048, but this is now [2] [2048] and cannot be combined since numbers cannot change between moves)
Good: 2048 (it's already one number)
Bad: 2047 (it's not a power of 2)
Bad: 11 (there are no 1's in the game)
Good: 000040000000 (zeroes are empty spaces)

Miscellaneous rules:

Input can be from anywhere reasonable, i.e. STDIN, function argument, file, etc.
Output can also be anywhere reasonable, i.e. STDOUT, function return value, file, etc.
Ignore the grid size - 22222222 should still output true.
The is no maximum to what s number could be, as long as it's a power of two. Therefore the possible numbers are any power of two greater than 0.
For those worried about zeroes causing ambiguity, that is not the case. For example, 22048 can be parsed as either [2] [2048] or [2] [2] [0] [4] [8]. The first one doesn't work, but the second one does, so it should output true.
This is code-golf, so the shortest code in bytes will win!

A:

Python: 457 422 characters
z=range
def c(n):
 for x in z(1,12): 
  if int(n)==2**x:return 1
 return 0
def p(s):
 if s=='':return[]
 for i in z(len(s),0,-1):
  if c(s[:i])>0and p(s[i:])!=1:return [int(s[:i])]+p(s[i:])
 return 1
def r(a):
 if len(a)==1:return 1
 i,t=1,a[:1]
 while i<len(a):
  if a[i]==t[-1]:t[-1]*=2
  else:t+=[a[i]]
  i+=1
 if len(t)==len(a):return 0
 return r(t) 
def f(s):
 if p(s)==1or r(p(s))==0:print('bad')
 else:print('good')

The function f(s) gets a string of digits and outputs 'good' or 'bad' accordingly.
I chose not to use 0 as spaces because spaces are meaningless in the game and they create ambiguity when parsing strings (is 22048 good or bad?). This only uses numbers up to 2048, but that can be changed without adding characters.
At the cost of 10 characters or so I can also print all the steps of combining the numbers.
And I realize that this code isn't golfed enough yet; don't worry, edits are coming.

A:

GolfScript, 137 characters
[0`]1$,4*,{2\)?`}%+:W;[]\]][]\{{:A;W{A 1=\?!},{[.A~@,>\@~+\].~!{0-.,/@+\0}*;}%~}%.}do+.{~}%,{{.-1%}%.&{[{.2$={+)}*}*]{1|(}%}%}*{,1=},!!

Input must be given on STDIN. The output is 0/1 for bad/good numbers. Most of the code is necessary for parsing the possible inputs.
This shorter version (113 characters) performs a simple shift test which would not work correctly for input like 224422.
[0`]1$,4*,{2\)?`}%+:W;[]\]][]\{{:A;W{A 1=\?!},{[.A~@,>\@~+\].~!{0-.,/@+\0}*;}%~}%.}do+{W{~[.:S]/[S.+]*}/,1=},!!

All test cases can be checked online.

