EFF: Unfortunately, we need long periods of data to do the relevant tests, and we do not have good long-term data on bond markets outside the U.S. For the U.S. we only have good long-term data (from CRSP) for Government bonds. 

Tests on the U.S. data do not indicate that the small average premiums observed in the returns on long-term versus short-term Government bonds are badly out of line with the predictions of asset pricing models. The models do not predict big differences in the returns on long-term and short-term governments, and the observed premiums are statistically consistent with the models. The same is true for the rather small premiums of corporate bond returns over Government bond returns. 

Having said this, there is another possibility that has bugged me since I first started to do research on bond returns about 35 years ago. The game for many active bond managers is forecasting moves in long-term interest rates. Long-term corporate bonds are not a good play in this game since they are typically callable, and they are likely to be called when long-term rates fall and long-term bond prices rise, robbing the interest rate timer of his gains. Long-term Government bonds have typically not been callable. This can create a demand from interest rate timers that raises the prices and lowers the expected returns on long Governments. This is not market inefficiency. It just says that there is something missing from the asset pricing models (the models for expected returns) that researchers commonly apply to bonds. As noted above, however, the available evidence does not provide strong support to this story.

KRF: While I certainly agree with Gene's answer, it is probably worth emphasizing another point Gene made more than 40 years ago. We cannot say anything about market efficiency without first saying what an efficient market is supposed to do. Or, more precisely, any test of market efficiency is simultaneously a test of our model of market equilibrium. So, is the absence of a meaningful premium for long-term U.S. bonds evidence of market inefficiency? Only if we are certain the premium is supposed to be positive, and it is easy to think of reasons why it might not be. For example, for pension funds trying to immunize their long-term (nominal) commitments or workers trying to lock in the nominal payoffs on their retirement savings, long-term bonds would be safer than short-term bonds. If the demand for long-term bonds as a hedge against variation in the interest rate is high enough, it could push the market's rationally required term premium to zero.