Analysis of RTs when speed-accuracy trade-off leading to ceiling accuracy has occurred
You may be able to reduce errors by telling participants to react slowly. However, by adding this instruction, you may also dilute the response time differences that your are trying to find (whether between or within subjects). If you are lucky, you are just adding random noise, however it is more likely that you will add confounds or tap into different mental processes (occuring at a later time). 

Another option to tackle this problem is to hold the speed more or less constant, by giving participants a very narrow time window to respond (which can also be tailored to their speed). As a result, they will produce more errors, which now become the dependent variable (see Draine &amp; Greenwald, 1998, who have introduced this technique).

References

Draine, S. C., &amp; Greenwald, A. G. (1998). Replicable unconscious semantic priming. Journal of Experimental Psychology. General, 127, 286–303.