Method -> {"EquationSimplification" -> "Residual"} in NDSolve helps.
n = 300;
m = 300;
k = 0.65;
eqns = Table[?[i]'[t] == RandomInteger[{0, 1}] - (k/n)*Sum[(1 + Cos[?[j][t]])*Sin[?[i][t]], {j, 1, n}], {i, 1, m}];
sol = First@NDSolve[{eqns, Table[?[i][0] == RandomReal[{0, 2}], {i, 1, m}]}, Table[?[i][t], {i, 1, m}], {t, 0, 500},
Method -> {"EquationSimplification" -> "Residual"}];
For example ,only one theta:
{?[299][t]} /. sol /. t -> 500
few theta's:
{?[1][t], ?[2][t], ?[3][t]} /. sol /. t -> 500
all theta's (n=m=300):
Table[?[w][t], {w, 1, n}] /. sol /. t -> 500
in Multicolumn format:
Multicolumn[Table[?[w][t], {w, 1, n}] /. sol /. t -> 500, 10, Frame -> All, Spacings -> {1, 1},
Background -> {Automatic, {{LightOrange, White}}}, Frame -> All, FrameStyle -> White, Alignment -> Center]