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 $n,p$: 

[![enter image description here][1]][1] 

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][2]. Note, in particular, the presence in that asymptotic expansion of the 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 is 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 $\{(n,g_{0.6}(n))\colon n=1,\dots,300\}$: 

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/7t13X.png
  [2]: https://link.springer.com/book/10.1007/978-3-642-65809-9
  [3]: https://i.sstatic.net/eLawl.png