This "odd behavior" is not peculiar to the negative binomial distribution.
A somewhat similar behavior is exhibited e.g. by the binomial distribution. For instance, here is the graph $\{(n,g_{0.6}(n))\colon n=1,\dots,300\}$, where $g_p(n):=P(X_{n,p}>EX_{n,p}+2\sqrt{Var\,X_{n,p}})$ and $X_{n,p}$ has the binomial distribution with parameters $n,p$:
The cause of such "quasi-periodic" behavior is the discreteness of the underlying distribution (say, that of $X_{1,p}$). This phenomenon is well explained by the asymptotic expansion (in powers of $1/\sqrt n$) of the (standardized) c.d.f. of the sum of lattice-valued i.i.d. random variables given, say, by Theorem 6 in Ch. VI of Petrov's book. Note, in particular, the presence in that asymptotic expansion of the $1$-periodic, somewhat trig-like "functions" $$S_{2k}(x):= \sum _{l=1}^{\infty } \frac{\cos (2 \pi l x)}{(2 \pi l)^{2 k}},\quad S_{2k+1}(x):= \sum _{l=1}^{\infty } \frac{\sin(2 \pi l x)}{(2 \pi l)^{2 k+1}}$$ (as well as the "function" $\delta_\nu$, which depends on $\nu\text{ mod }4$). Here are the graphs $\{(x,S_{2k+1}(x))\colon-1<x<2\}$ for $k=0$ (left), $k=1$ (center), and $k=2$ (right):
Here is also the graph $\{(n,S_1(2 \sqrt{n p (1-p)}+p n+\lfloor -p n\rfloor)) \colon n=1,\dots,100\}$ for $p=0.6$, relevant to the above graph $\{(n,g_{0.6}(n))\colon n=1,\dots,300\}$:
