TITLE: Why is the string group not a Lie group?
QUESTION [9 upvotes]: The string group $String(n)$ is by definition a 3-connected cover of $Spin(n)$. This definition determines the homotopy type of the string group.
[In a previous version of this question I screwed up the definition and caused some confusion, see the comments below.]
A common argument is saying that "the string group cannot be a Lie group because it has vanishing $\pi_3$". This is obviously not a complete argument because $(\mathbb{R},+)$ is a nice Lie group with vanishing $\pi_3$. 
What is the correct statement about Lie group structures on the string group, and how does one prove it?

REPLY [9 votes]: To follow up, there is now an infinite-dimensional Lie group model of String:

Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. 16 (2013) 3678-3721, doi:10.1093/imrn/rns154, arXiv:1104.4288.