Because finite automata are, in general, strong abstractions of systems, many test cases, which are automata traces generated uniformly at random, may be unconcretizable. This paper proposes a method extending the aforementioned testing approach to pushdown systems providing finer abstractions. Using combinatorial techniques guarantee the uniformity of generated traces. In addition, to improve the quality of the test suites, the combination of coverage criteria with random testing is investigated. The method is illustrated within both structural and model-based testing contexts.