Some tests might test if there exist deterministic trend which makes series non-stationary in itself but stationary around trend.
Some techniques which I have used includes:
1) Autocorrelation function. If autocorrelation shape is such that decline is very slow and autocorrelation at lag 1 is close to unity then it means that series depends greatly on previous value.
2) Spectral density function. If most of power is in low frequencies, then it means that series is driven by long-run trends.
3) Eye-ball regression, just visually examine series.
My statistics professor used to say that we economists are often concentrated to study only certain kind of theoretical models and do not allow other forms of non-stationarity. And this might make it difficult to apply these models for real world processes! :)