I recently came across a trinitarian argument demanding that Psalm 22 be a Messianic psalm, which it is not. Viewing it as such the trinitarian focused upon one verse:

Psalm 22:10 Upon You I was cast from birth; You have been my God from my mother’s womb.

The argument found Jesus being God in his preexistence by lacking one as his own God, which he only came to have upon becoming a man.  Yet as many Trinitarian commentators recognize, the psalm was not messianic.

To be clear this is not to say that parts of the psalm did not find a messianic application, but finding such in key portions is vastly different than having the entire psalm attributed to the Messiah.  So The Interpreter’s One-Volume Commentary (p. 268):

In this overpowering sense of alienation, shared by Jesus on the cross (Matt. 27:46; Mark 15:34), the psalmist draws comfort from reflection on the history of his people.

The above reference clearly distinguishes between the psalmist to whom the psalm applied and Jesus as one who found a later fulfillment of the specific text.  Such does not indicate that the text originally applied to Jesus or that the entire context related to him (cf. 2Sa. 7:14; Heb. 1:5).