Slawek, I don't formally support Lightning Components yet, but you can develop them if you don't mind some workarounds for the moment. I've resolved this as a duplicate of #49 where those workarounds and issues are being actively discussed. For your convenience, here's the most relevant comment about the issues you're having:
Based on my own experience playing with lightning components in and outside of IC, you cannot deploy only part of an aura bundle successfully, e.g., just a .css/.cmp/.js file. Best case scenario, it will deploy successfully but effectively replace the existing bundle with one containing only the deployed file, but more often it just bombs out. The workaround until I add formal support for lightning components is to deploy the entire aura bundle and not the single file. The easiest way to do this is to use Deploy All Metadata and then do a Custom selection on the bundle directory itself.
Scott, there is a possibility to deploy only part of an aura bundle and in MavensMate v7(beta) it is implemented.
I hope You will start support Lightning Components soon, because IC is great in every other way. Thank You for fast answer and I keep my fingers crossed for you.
Scott Wellsrepo owner
Slawek, thanks for the kind words. First class support for lighting components is the next item on my backlog. My guess is that I'll have something in January, but I might leak out some early work for those already developing in lightning. Partial deployment will definitely be part of that work.