What do you mean by "bolt action rifle" and what do you mean by "machine gun"? I know, I know, it is getting nitpicky, but bear with me. In this old post I go into much more detail on the development of the Breechloader, so I would suggest you read that, as I'm going to quickly summarize what I wrote there. The important key points are that a bolt mechanism was first introduced in the 1840s, but it might not be what you're thinking of. The Dreyse "Needle Gun" was a single shot rifle. Each round was loaded manually by hand. We don't see a repeating bolt action rifle in general military service, which might be what you are imagining in your head, until 1869, when the Swiss adopt the Vetterli (it had been in development a few years by then, and the 1867 model exists, but it was not adopted for general use). From there design improvements focus especially on the magazine. The Vetterli used a tubular design, which was outdated once James Paris Lee brought the box magazine around a decade later, and Mauser introduced stripper clips and staggered magazines, and by the early 1890s, the core bolt action rifle design has been finalized, and we just argue about which one is best without there being a clear winner.

Now, as for machine guns, well... again... what do you mean? If you're picturing a Gatling Gun, that dates to the early 1860s, so before the Vetterli. But, the Gatling Gun isn't actually a machine gun, as a true machine gun doesn't require you to do anything except pull the trigger, and it then continues to fire as long as you hold down the trigger. The Gatling Gun was crank operated with a number of barrels, basically a machine that allowed a person to quickly load and fire several connected single-shot guns. It would continue to fire as long as you kept turning that crank, but it doesn't fit the actual definition of a machine gun. For that we have to wait two more decades, with Hiram Maxim's eponymous Maxim Gun in the 1880s, which used its own recoil to continue firing while the trigger was depressed, rather than requiring manual labor on the part of the gunner.

So there you have it: Bolt action mechanism came first, then came the rapid-fire, but not actually a machine gun Gatling Gun, then came the repeating bolt-action rifle, and THEN came the true machine gun. I would also add that I didn't say anything about non-bolt action repeating firearms, but lever-action repeaters and revolvers came before the Gatling Gun.