  On the one hand, we provide the first examples of arbitrarily highly
connected (compact) bad orbifolds. On the other hand, we show that n-connected
n-orbifolds are manifolds. The latter improves the best previously known bound
of Lytchak by roughly a factor of 2. For compact orbifolds and in most
dimensions we prove slightly better bounds. We obtain sharp results up to
dimension 5.
