I don't think there's a maximum SNR or SINR per se. SNR could be infinite, in principle, and the receiver should work great. The problem will be saturation of the receiver, with more than so much signal level. Or another way to say it is, if the receiver becomes overwhelmed with input signal, it creates intermodulation distortion products, and therefore the very high input power actually reduces SINR, inside the receiver.
Here's an interesting presentation. The receiver design they show becomes saturated at a power level of -10 dBm, at the receiver input.
The SINR you show in your table is the ratio S / (N + I), at the receiver. What the table says is that if this ratio is as low as -6.7 dB, you can still successfully receive the signal, but only if you use QPSK modulation, and a strong FEC, which results in low spectral efficiency.
At the opposite extreme, if you use 64-QAM and a much weaker FEC, you can achieve a fairly impressive spectral efficiency, but the price you pay is that the channel has to be very clean. The marginal SINR is now 22.7 dB, which is quite benign.
So, how does one typically achieve a high SNR at the receiver? Since there isn't much you can do about noise in the propagation channel and in the receiver electronics, all you can do is increase signal power at the transmitter, increase the gain of the transmit antenna, and/or increase the gain of the receive antenna.
You asked what the highest SNR or SINR is. If you conceive of the noise level being very low, SNR or SINR could theoretically be extremely high. That's not really the point. The point is that you have limits to how much transmit power is practical, *and* you have limits in how much input signal power the receiver can cope with. You cannot just keep increasing input signal power at the receiver indefinitely. The link I showed you gives an example of what happens when the receiver becomes overloaded, in their case, with -10dBm of signal strength.
So you need to rephrase your question. Marginal SNR or SINR is always the lowest possible figure required for successful demodulation of a signal. The highest SNR or SINR is in principle infinite (meaning, zero noise, zero interference, even with a practical amount of transmitter power).
But there *is* a "maximum" you should be concerned with. It's not the maximum SNR. Instead, it could be the maximum signal power at the receiver's antenna connection, which depends on receiver design. Another maximum level that would be involved here is the maximum allowable transmitter power, which would normally be limited by human health concerns, especially with cellular communications, where the transmitters are all around us (cell phones and cell towers).
Over the top SNIR makes little sense in a networked system. Every extra dB is only increasing interferences to other users, and is also consuming more energy, so overall you only lose. If I can define a maximum as a requirement of a maximum rate modulation scheme, then yes, there is such thing as maximum SNIR.