Is it possible to analyze survey data with two or more levels of clustering with the svy commands?

The svy commands actually allow multiple levels of clustering, but you only need to specify the first level.

This is because the variance estimator used by the svy commands allows any amount of correlation or clustering within the primary sampling units. See [U] 30.2.1 Multistage sample designs.

This method (to only account explicitly for the primary sampling units) is a standard one in survey statistics.

There are other variance estimation methods that explicitly account for secondary sampling. However, these methods require more assumptions than the svy estimator. If these assumptions are correct, then these other methods may yield more efficient variance estimates. However, the svy estimator is more robust than these methods.

Thus, most survey statisticians use the svy variance estimator or other methods, such as a jackknife, that only account explicitly for the primary sampling units.