The solution components all have size around 1. The norm of the error is also around 1. That gives some indication that there are at most one digit or so correct.

Were you not asking about $\|x^\ast-\bar{x}\|$? Norm[x]is just the Mathematica expression for $\|\mathtt x\|$.

Well, you could just print the intermediate results. That might make clear what are the vectors in use and what that norm is from.

In[53]:= approxsol = LinearSolve[mat, bv]

Out[53]= {1.01850881228, 0.783151176337, 1.53552721482, 0.645697515412}

In[54]:= diff = approxsol - sol

Out[54]= {0.0185088122791, -0.216848823663, 0.53552721482, \
-0.354302484588}

In[55]:= Norm[diff]

Out[55]= 0.678001207119

Numerically this is garbage anyway because the use of Round has made alterations a few digits in that, based on the conditioning, really should throw the result away from sol. To avoid this one might instead do as below.

mat = N[hm, 5]
bv = N[hm.sol, 5]

(* Out[56]= {{1.0000, 0.50000, 0.33333, 0.25000}, {0.50000, 0.33333, 
  0.25000, 0.20000}, {0.33333, 0.25000, 0.20000, 0.16667}, {0.25000, 
  0.20000, 0.16667, 0.14286}}

Out[57]= {2.0833, 1.2833, 0.95000, 0.75952} *)

approxsol = LinearSolve[mat, bv]

(* Out[47]= {1.0000, 1.0000, 1.000, 1.0000} *)

I should mention that, as on Mathematica.stackexchange.com, I was unable to make any sense of the initial paragraph in this query. Also it would be helpful to provide a link when cross-posting essentially the same query from a different forum.