The nearest-neighbor (NN) heuristic (among others) is analyzed in this paper:

Johnson, David S., and Lyle A. McGeoch. "The traveling salesman problem: A case study in local optimization." *Local search in combinatorial optimization* 1 (1997): 215-310.
(PDF download link.)

They say:

In the specific case of "random Euclidean instances"
(in contrast to "random distance matrices"),
they observe experimentally a fixed ~25% longer path length, which
would exceed any fixed $n$ for large enough nodes $N$.
In terms of both theory and experiments with random distance matrices,
the growth rate is $\log N$.
Either way, any fixed $n$ could be exceeded with sufficiently large $N$,
so the answer to the OP's question is

*Yes*.

The cited paper is:
Rosenkrantz, Daniel J., Richard E. Stearns, and Philip M. Lewis, II. "An analysis of several heuristics for the traveling salesman problem." *SIAM Journal on Computing* 6.3 (1977): 563-581.
(Journal link.)

**Added** (in response to question from Manfred).
Here is the essence of the Rosenkrantz et al. lower bound:

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(Slide from David Johnson PowerPoint Lecture.)
}

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