I know this sounds weird, but contamination of milk *may* be possible *in the goat*, according to some papers I've read (and subsequently lost). In fact, when they are doing experiments which require sterile, but not treated milk, they have to be careful about the feed of the animals because it is thought that bacteria can enter the system through that route. Unfortunately no known mechanism allows that to happen so it's kind of up in the air right now.
The main thing I want to communicate, though, is that milk contamination can come from sources *other than your handling*. There may be systemic issues beyond your control. In fact, I remember seeing a study of raw milk contamination issues and it was saying that there is a reverse correlation between the quality of handling and incidence of contamination (i.e. the most controlled milk producers have the highest incidence of contamination). This was in the US and UK *only*, though, and I think the implication was that standards are so high that larger, more automated systems tend to produce contamination because single issues can become more systematically problematic as opposed to smaller producers that tend to vary their approach more often.
I think there is a tendency to assume that because you are doing everything right, that there is basically no chance of a problem, but that just doesn't seem to be the case in milk production. I wish I wasn't so lazy about keeping references for these kinds of things :-( But it's worth reading as much current research as you can because understanding is changing in this space pretty quickly.