The space you get through such a process of cell attachments is at least homotopy equivalent to a CW complex, although maybe not homeomorphic (as Tyler's example shows).
This comes up in Milnor's Morse Theory (see pp. 21-24), where you build a CW complex inductively by attaching cells according to the critical points of a Morse function on a manifold. There's no reason that the indices have to be increasing, and this causes one to attach cells "in the wrong order," i.e. a cell of dimension n may be attached to existing cells of dimension n or greater. In his book on Morse Theory, Milnor shows that up to homotopy the resulting space is a CW complex. The basic point is that each attaching map can be deformed to the appropriate skeleton, and attaching a cell along two homotopic maps produces two homotopy equivalent spaces (this is a bit tricky, and Milnor attributes it to Hilton). Finally, an argument with homotopy colimits allows one to treat the case of countably many cell attachments.
I suppose that this applies to Tyler's example, and shows that the space he builds is homotopy equivalent to an infinite wedge of circles.