I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?
Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that
an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the option market in extreme strike regions and indicate that implied volatilities ﬂatten out and converge asymptotically to a constant. This behavior is not consistent with the rapidly diverging asymptotics which are implied by SABR.
The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.