The word "Testament" is the Latin form of the word for "covenant". God made several covenants with his people and promised in Jeremiah 31:31 to make a new covenant with Judah and Israel in the future. The writings that we call the Old and New Testaments are not the same as these covenants and it's misleading to call them that.
Probably more serious is the fact that calling a collection of writings the "Old Testament" implies that it has been superceded and no longer has any validity. This is the language of replacement theology which offers the view that God rejected Israel as his people (even though he promised he never would) and replaced them with the Christian church. It is this view that has led to so much anti-Semitism and bloodshed through the centuries.
Jesus referred to these writings as the Law and the Prophets, or the Law, the Prophets and the Psalms. Jewish people today use the same three-fold division of the Scriptures, the Law, the Prophets, and the Writings (the first and largest of which is Psalms). They typically use an acrostic, the word "Tanakh", which is made up of the first letter of Torah (law), Nevi'im (prophets), and Ketuvim (writings). Another alternative is to call it the Hebrew Bible or Hebrew Scriptures.