What's wrong with the Sum function?

I can confirm that.

Testcode: (The sums are giving out different identities (the second one is wrong) while NSum works as expected.)

a=Sum[1,{K,0,n},{M,0,n-K},{La,0,K},{Lb,0,n-K-M},{Lc,0,M}]
b=Sum[1,{La,0,n},{Lb,0,n-La},{Lc,0,n-La-Lb},{K,La,n-Lb},{M,Lc,n-K-Lb}]
TrueQ[FullSimplify[a==b]]
({a,b}/.n->#)&/@Range[1,10]
{NSum[1,{K,0,#},{M,0,#-K},{La,0,K},{Lb,0,#-K-M},{Lc,0,M}],NSum[1,{La,0,#},{Lb,0,#-La},{Lc,0,#-La-Lb},{K,La,#-Lb},{M,Lc,#-K-Lb}]}&/@Range[1,10]

Seems to be a bug in Sum.

$Version

11.0.0 for Microsoft Windows (64-bit) (July 28, 2016)

Also confirmed by my other machine:

10.4.0 for Microsoft Windows (64-bit) (February 26, 2016)

$Version

"11.0.1 for Mac OS X x86 (64-bit) (September 21, 2016)"

Clear[f, g]

f[n_] := Sum[
  1, {K, 0, n}, {M, 0, n - K}, {La, 0, K}, {Lb, 0, n - K - M}, {Lc, 0, M}]

g[n_] := Sum[
  1, {La, 0, n}, {Lb, 0, n - La}, {Lc, 0, n - La - Lb}, {K, La, n - Lb}, {M, 
   Lc, n - K - Lb}]

The symbolic closed-form functions are not equivalent.

f[n] // FullSimplify

(*  1/120 (1 + n) (2 + n) (3 + n) (4 + n) (5 + n)  *)

g[n] // Simplify

(*  1/12 (1 + n) (2 + n)^2 (3 + n)  *)

Solve[f[n] == g[n], n]

(*  {{n -> -3}, {n -> -2}, {n -> -1}, {n -> 0}, {n -> 1}}  *)

However, FindSequenceFunction applied to sequences generated from either f or g give the symbolic closed-form function for f

f[n] == FindSequenceFunction[f /@ Range[7], n] // Simplify

(*  True  *)

f[n] == FindSequenceFunction[g /@ Range[7], n] // Simplify

(*  True  *)