I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense.

Given a real number $t \geq 2$, call $P(t)$ the following assertion.

"There exists $N \in \mathbb{N}$ such that $\sum_{i=1}^{[n/t]} \binom{n}{i} \leq \binom{n}{[n/t]+1}$ for every $n \geq N$".

Here $[a]$ denotes the floor of $a$ (the largest integer at most $a$). Note that $P(t)$ is clearly false if $t \leq 2$. Using what I found here (answer of Michael Lugo):

Sum of 'the first k' binomial coefficients for fixed n

I was able to show that $P(3)$ is true.

My question is: is it true that $P(t)$ is false for every $2 < t < 3$?