Linear regression of nonstationary variables? Ok, so I'm quite new to this topic. Based on what I've read, running a regression on nonstationary variables can give you nonsense.
But what if you want to show that two "bounded" nonstationary series (I'm not sure if they are nonstationary if they are bounded?) go "hand-in-hand", i.e. that the level of X is high when the level of Y is high and vice versa? What would you do?
 A: You will have to use (as @Cagdas Ozgenc also mentioned) a technique called co-integration.  If the time series $y_{1t}$ and $y_{2t}$ are non-stationary, then they are said to be co-integrated if there exists a linear combination of the two series that is stationary.  
There are several references for it like e.g. Hamilton, J.D. (1994). Time Series Analysis, Princeton Unversity, Press, New Jersey or Johansen, S. (1995). Likelihood Based Inference in Cointegrated Vector Error Correction Models, Oxford University Press, Oxford.
The idea is simple, for the linear reagression model $y=\beta_0 + \beta_1 x + \epsilon$ where the usual assumptions are fullfilled you see that $(y  - \beta_0 - \beta_1 x) \sim N(0;\sigma)$ (because it is $\epsilon$), so with OLS you find a linear combination that is 'white noice' or 'stationary'. 
The idea behind co-integration is to find, for non-stationary time series, a linear combination that is stationary, so for $y_{1t},y_{2t}$ non-stationary, you will try to find a linear combination $\gamma_1 y_{1t} + \gamma_2 y_{2t} \sim I(0)$ and you see the similarity with the OLS equation.  
When you find the linear combination, then you have found that $y_{1t} = \frac{\gamma_2}{\gamma_1} y_{2t} + \delta$ where $\delta$ is $I(0)$, or (up to a term that is 'white noice') $y_{1t}$ and $y_{2t}$ 'move together'. Co-integration is about finding that linear combination.  
Obviously for co-integration OLS will not work. 
A: Both bounded and non-bounded series could be stationary or non-stationary. In simple language, stationarity means "doesn't change over time". Clearly bounded variables could change over time. You can change any continuous time related variable to a bounded one by changing it to % of maximum. So, suppose you regressed "Dow Jones Industrial Average" (as a % of 20,000) to "Population of China" (divided by 8 billion). There would be a strong relationship.
One solution is to remove the non-stationarity before doing the regression. 
