The answer to the question in the Bredon Problem is given in my paper ,,Group actions with inequivalent representations at fixed points“, Math. Zeitschrift, Vol. 187(1984), pages 29–47.

It turns out that for a compact Lie group G, the answer depends on the quotient group G/G* of G, where G* is the connected component of the identity element of G.

Theorem. Let G be a compact Lie group such that G is connected or each element of G/G* is of prime power order. Then, for any smooth action of G on an acyclic manifold and for any two fixed points x and y, the G-representations determined on the tangent spaces at x and y are isomorphic to each other. The same conclusion holds for any smooth action of G on a homology sphere with at least three fixed points.

Corollary. Let G be a compact Lie group such that G is connected or each element of G/G* is of prime power order. Then, for any smooth action of G on any acyclic manifold (resp., homology sphere), any two connected componets of the fixed point set F are of the same dimension. 

Theorem. Let G be a compact Lie group such that G/G* has an element not of prime power order. Then there exists a smooth action of G on some Euclidean space (resp., disk; sphere) such that the fixed point set F has two connected components of distinct dimensions.

Corollary. Let G be a compact Lie group. Then, for every smooth action of G on some Euclidean space (resp., disk; sphere), the connected  components of the fixed point set F all are of the same dimension if and only if each element of G/G* is of prime power order. 

The following problem goes back G. E. Bredon, “Introduction to compact transformation groups”, Academic Press, 1972, page 58.

Bredon Problem. If a compact Lie group G acts on
a disk, sphere, or Euclidean space, is it true that the fixed point set F always consists of connected components of the same dimension?