Johansen's $\Pi$ is full rank except variables are non-stationary I have two variables. They're both $I(1)$ even when I fit constant and trend terms into the ADF test. The $p$-values for the stationarity tests are around 0.5 so it's not a marginal case.
However, when I execute the Johansen procedure with a constant in the error correction term the $\Pi$ matrix is full rank by the trace and maximum eigenvalue tests ($p < 0.01$). I've read a textbook and a paper on the consequences of $\Pi$ being full rank  and they both state that this implies that both variables in the system are $I(0)$. It's important to note that if I exclude a constant from the ECT or if I add a deterministic time trend, then the rank of $\Pi$ is estimated as 1 (instead of full rank).
Is this most likely a type 1 error or could there be more going on here (such as partial integration)?
 A: You have answered your own question. If you get cointegration when adding trend, or removing constant, this means that these terms are important, especially when without these terms you get conflicting results. To test the intuition I suggest doing Monte-Carlo simulation with simple unit roots. Also if you have two variables you can also test cointegration using Engle-Granger approach.
A: If two variables are I(1) but are not co-integrated, it simply means that there is no long run-relationship between the two variables. However, there may be short run relationship. So you need to run the VAR in first difference. If two variables are I(0),we can always use OLS. There is no justification to use co-integration.  
Johansen’s co-integration model describes five different cases regarding the intercept and trend in the co-integrating equation. For details refer to the paper by Ahking (2002) in Journal of Macroeconomics. The results will differ but which to choose depends upon the data and the theory.
