The Shulhan Aruch (Orah Haim 42:3) writes that if one keeps his Tefillin in a bag which is exclusively designated for Tefillin, then once he placed his Tefillin in the bag, he may not then use the bag to keep money inside it. The bag at that point becomes a "Tashmish Kedusha" – an article designated for the purpose of a sacred article – and thus may not be used for money. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) clarifies that the Shulhan Aruch mentions money only as a common example, and in truth, this Halacha applies to all items which are not sacred like Tefillin. Even things which are used for a Misva – such as a Siddur or Tehillim – may not be placed in the bag designated for Tefillin. Likewise, one may not keep in this bag the mirror he uses to help position the Tefillin on his head, or the marker he uses to paint his Tefillin.
Many people keep their small Tefillin bag in a larger bag, which is used also for their Tallit and Siddur. Since this larger bag is not designated exclusively for Tefillin, it may be used for other items, even items which are not used for a Misva, such as money. This Halacha is discussed by Hacham David Yosef (contemporary) in his Halacha Berura (vol. 3, p. 301; listen to audio recording for precise citation).
Summary: A bag which is designated exclusively for Tefillin may not be used to hold anything else, even a Siddur. However, if one keeps his Tefillin bag in a larger bag together with his Tallit and Siddur, then since the larger bag is not designated exclusively for Tefillin, he may keep anything else – even money – in that larger bag.
