I think Nim absolutely has a place competing with Rust too.
It has several tunable GCs allowing for great flexibility there, and also supports running with no GC at all. It compiles to C that will compile nearly anywhere C does. It produces smaller binaries, and it can easily link against musl instead of libc for an even lighter footprint. It's also safe like Rust is, and the compiler will help you catch a ton of errors.
That's not to say that it's competitive at this point in time. It's much more unstable as a language due to not reaching 1.0 yet. They only just (August) received any real kind of funding, allowing them to hire on a new developer but it's still just a handful of core devs and some dedicated contributors. The standard library and the community repo (Nimble) can't compare with Rust's and Cargo in terms of package availability and support.
BTW, I would see Nim more as a competitor to D, Swift and Java than to Rust.