I used SU6656 and it shows an increase of phosphorylation of the src family kinase. Is it possible to increase phosphorylation of the target protein by an ATP-competitive inhibitor..?
Yes, it can happen and it's most likely due to negative feedback regulation of signalling pathways.
If a negative feedback dependent on the target protein occurs, in the presence of the inhibitor such negative feedback response will be blocked. Thus, after treatment of the inhibitor, one might observe an increase in the phosphorylation of the target protein because the signalling pathway upstream of it will remain active.
To certify that the inhibitor is working, I recommend to check the phosphorylation of a substrate of the target protein, which is expected to be reduced or undetectable in the presence of the inhibitor.