Byzantine generals problem with signed messages (Lamport)
Maybe this is what confuses you -- mixing the byzantine agreement problem (broadcast: where there is one transmitter and multiple receivers) and the byzantine consensus (where every party starts with an input, and if all have the same input, this should also be their output). The bound of at most half traitors stems from the consensus requirement.

Say we have $n$ generals, out of which $t$ are traitors and $\ell=n-t$ are loyal. Assume general $i$ holds the input bit $b_i$

Now assume they all perform a distributed majority vote protocol. Is the protocol guaranteed to succeed?
This questions is actually not a well defined. The success of the protocol can be interpreted in two different ways:


All loyal generals output the same answer: $MAJ ( b_1, \ldots, b_n)$
All loyal generals output the same answer: $MAJ (L)$ where $L=\{b_i \mid $ general $i$ is loyal$\}$.


Note that using Lamport's result you mentioned, if $n&gt;t+2$ then all the loyal parties are guaranteed to output the same output even if the traitors misbehave and don't follow the protocol, but which output do they output? Assume $\ell=3$ as you suggest, and assume they all hold the input $0$. The "traitors" can still force the loyal generals to output "1": they just set their input to $1$ and behave honestly! Remember that the loyal parties don't know who is loyal and who isn't (or even if there are traitors at all -- maybe everyone is loyal). From their point of view, there are $t$ votes for "1" and only 3 votes for "0", and since everybody behaves as supposed by the protocol -- the output must be 1! This suits interpretation number (1) above.

Thus, interpretation number (2) above makes sense only for the case where  setting the input of the traitors does not affect the majority. 
A special case is the consensus problem: if all the $\ell$ parties hold (say) "0", we want to guarantee that they all outputs a 0. Of course, if $t&gt;\ell$ then the simple "attack" described above will work. So we need $t&lt;\ell$, or equivalently, $$n=t+\ell \ge t+(t+1)=2t+1.$$

In the most general case of byzantine major-vote, 
the loyal parties may be split
(e.g., $\lfloor \ell/2\rfloor$ votes vs. $\lceil \ell/2\rceil$ votes). In this case, even a single traitor (playing honestly) can determine the result by himself.