Which compact groups have nonisomorphic irreducible representations of the same dimension? If $\Gamma$ is a compact simply-connected semisimple Lie group, then the Weyl Dimension Formula tells us exactly which dimensions it can act irreducibly on.
For certain $\Gamma$, it is easy to find pairs of nonisomorphic representations of the same dimension:
1). $A_n (n\geq 2)$, $C_n$ ($n$ = A116940(k)), $D_n (n\geq 4)$, and $E_6$ each have pairs of fundamental irreducible representations of the same dimension.
2). Additionally $G_2$ has two irreducible representations of  dimension 77.
Furthermore, given that $\Gamma$ has one pair of nonisomorphic representations of the same dimension, it is easy to prove (using the Weyl Dimension Formula) that it has infinitely many such pairs.
Question 1: Among the remaining groups not mentioned above, which are known to have pairs of nonisomorphic irreducible representations of the same dimension?
Question 2: For obvious reasons $A_1$ cannot have such pairs, but are there any other cases where one can rule out the existence of such pairs?
It would seem that since the Dimension Formula so greatly restricts the possible dimensions for the other groups that in the long run there must be pairs of irreducibles of the same dimension; if anyone knows of any results along the lines of such heuristic forcing arguments those would be useful as well.
Edit: Robert's answer below reminded me that such pairs also occur for:
3). $B_n$ when $n$ = A116940(k) just as in the $C_n$ case because of a result I proved awhile back. More generally, one can show that if $B_n$ has such a pair then so does $C_n$ and conversely if $C_n$ has such a pair then so does $B_n$, so Robert's comment also shows that $C_4$ and $C_5$ have such pairs (the pairs arising from Robert's examples are in dimensions 11354112 and 38928384 for $C_4$ and 24741150720 for $C_5$).
4). $F_4$ has two irreducible representations of dimension 1053 that I had completely forgotten about in my list.
 A: We give a table for irreducible representations up to dimension $2^{15}$ in the supplement of our preprint http://arxiv.org/abs/1012.5256v1 
I immediately found the following examples with respect to the first question:
(1) $B_3$ occurs twice in dimension 112, 168, etc. 
(2) $B_4$ occurs three times in dimension 2772, twice in dimension 9504, etc. 
(3) $B_5$ occurs twice in dimension 23595. 
(4) $F_4$ occurs twice in dimension 1053.
A: Without offering a complete answer to the stated question, I'd first ask what significance the answer would have (one way or the other) in terms of Lie theory?  I'd also want to extract the essential numerical problem, which only concerns the numerator polynomial in Weyl's dimension formula.  (The denominator is constant and doesn't affect the outcome.)   
Given an irreducible root system belonging to a simple Lie type of rank $r$, the polynomial in the numerator is a product of $n$ factors each of which is a $\mathbb{Z}$-linear combination of $r$ variables $x_1, \dots, x_r$ with positive coefficients depending on the root system.   Here $n$ is the number of positive roots.  The question is when if ever this polynomial can take the same value at two different $r$-tuples of strictly positive integers.   If it does take the same value twice, it will take the same value infinitely many times: multiply each $r$-tuple by the same positive integer.   
From the viewpoint of root systems (compact groups being far in the background now), types $A_r$ with $r>1$, $D_r$ with $r>3$, and $E_6$ have graph automorphisms which guarantee a positive answer to the question.   Beyond this I can only see a possibility of accidental positive (or negative) answers as in the case of $G_2$.   By now the question is only remotely connected to Lie theory, via the fixed linear combinations of the variables occurring as factors in the given polynomial.    
