I think I can see an example using decomposition theory. I am mostly using the notation from [the wikipedia article on the Whitehead manifold][1]. I am not sure yet, whether it is an example for $n=3$ or $n=4$. Take $K_1=S^1\times D^2$ and $K_2=(S^1\times D^2) / W$ where $W=\bigcap_n W_i$ is the intersection of a sequence of nested solid tori such that each $W_{i+1}$ is embedded in $W_i$ as a Whitehead-link and we use the standard framing (although I am  not sure whether this makes a difference for the argument). 

Certainly $S^1\times D^2\times \mathbb{R}$ can be embedded into $\mathbb{R}^4$. Thus knowing that it is homeomorphic to $K_1\times \mathbb{R}$ would show that $K_1$ embeds into $\mathbb{R}^4$. I don't know whether it embeds into $\mathbb{R}^3$.

I believe that Bing shrinking can be used to show this and that I heard about this example in a lecture quite some time ago. I will try to find my notes or a reference or expand on my answer later. 


  [1]: https://en.wikipedia.org/wiki/Whitehead_manifold