Regarding NTRU homomorphic properties
Since you referring to the LWE variant of Stehlé and Steinfeld, I will try to give you an answer to your question in that context. Note that these results are a mere extension of the correctness condition in Lemma 3.7 from the revised version of the paper [SS13]. As I said in the comments, at the end it all depends on the choice of parameters ($n, \alpha, q$, etc), and additionally, on the number of additions you want to support. 

In this scheme, ciphertexts are of the form $c = h s + p e + m$, where $h$ is the public key, $s$ and $e$ are sampled from a Gaussian distribution, $p$ is a public parameter, and $m$ is the message. Public keys are of the form $h = p g f^{-1}$, where $f$ is the secret key, and $g$ a random polynomial. 

The correctness condition for one ciphertext is $||f \cdot c_i ||_{\infty} \leq q/2$. We want to check the conditions for decrypting correctly a sum of $k$ ciphertexts:

$c_{sum} = \sum\limits^k_{i=1} c_i = \sum\limits^k_{i=1}(h s_i + p e_i + m_i)$

Since $f h = f (p g f^{-1}) = p g$, then $f \cdot c_i = f h s_i + p f e + f M = p g s_i + p f e + f M$

Hence, we want to check whether $||f \cdot c_{sum} ||_{\infty} \leq q/2$, where:

$f \cdot c_{sum} = \sum\limits^k_{i=1}(p g s_i + p f e_i + f m_i) = \sum\limits^k_{i=1}p g s_i + \sum\limits^k_{i=1}p f e_i + \sum\limits^k_{i=1}f m_i$

Lemma 3.7 provides the following bounds:

$||pgs_i||_{\infty}, ||pfe_i||_{\infty} \leq 8 \alpha q n^{0.25} \omega(\log n)||p||^2 \sigma$

$||fM||_{\infty} \leq 4n||p||^2 \sigma$

Therefore, $||f \cdot c_{sum}||_{\infty} \leq 2k\cdot(8 \alpha q n^{0.25} \omega(\log n)||p||^2 \sigma) + 4n||p||^2 \sigma$. 

Next, assuming that $n^{0.75} \leq \alpha q$, as in Lemma 3.7, we have that: 

$||f \cdot c_{sum}||_{\infty} \leq 16 k \alpha q n^{0.25} \omega(\log n)||p||^2 \sigma + 4n^{0.25}\alpha q||p||^2 \sigma$

After some simplifications, we obtain the following bound:

$||f \cdot c_{sum}||_{\infty} \leq (16 k + 4) n^{0.25}\alpha q\omega(\log n)||p||^2 \sigma$

Note that when $k=1$, it produces precisely the same result than Lemma 3.7. Finally, similarly to Lemma 3.7, we have to initially assume that $ \omega( n^{0.25}\log n)\alpha q k ||p||^2 \sigma &lt; 1$, so the correctness condition for the sum of $k$ ciphertexts holds.

A final note: these results, as the one from Lemma 3.7, are not tight, and depend heavily on the choice of parameters. My guess is that you can sum (much) more than $k$ ciphertexts, and still decrypt correctly. 

References

[SS13] Stehlé, D., &amp; Steinfeld, R. (2013). Making NTRUEncrypt and NTRUSign as Secure as Standard Worst-Case Problems over Ideal Lattices. IACR Cryptology ePrint Archive, 2013, 4.