My favorite kind of answer to my favorite kind of question:
It depends
Some mods, like mods regarding high-res textures, are single-player mods that do not affect the communication with the server, and can therefore be used irrespective of server configuration.
Other mods, like aircraft mods, require server support in order to work.
In general, the description of any given mod will list if it's compatible with vanilla servers, if it only takes effect in SSP, or if it requires a modded server.
Edit: In response to comments; one piece of clarification:
Some mods function somewhat more like exploits based on the fact that the server allows a certain amount of client-side processing of things like falling. Mods that function by misrepresenting the effects of gravity, for instance, can function without server support, at least until the communication protocol is changed to handle all movement server-side, assuming that ever happens.
It's not possible to break bedrock with any amount of TNT, since bedrock has such a high blast resistance that TNT simply can barely make a dent in it, let alone overcome it completely.
Bedrock has a blast resistance of 18,000,000. (That's 3,000 times more durable than obsidian.)
The explosion math is not straightforward, but a great simplification is that a block is destroyed if its blast resistance is overcome by the explosion power applied to it. An explosion's damage is done by radiating a number of lines out from the centre of the blast, and applying damage alone those lines to blocks at intervals (every 0.3 metres) along the lines. If all of a line's damage is absorbed by one or more blocks it has passed through, no farther intervals are checked.
So, the amount of damage a block of TNT can do simultaneously to a particular piece of bedrock is the sum of the damage applied by all the rays that intersect it. This calculation has already been worked on that page as "The minimum block resistance required to absorb maximum blast force of an explosion happening in nearby air". For TNT under optimal conditions (the TNT is sitting on the block), this is 77.67. That means that a block of TNT will destroy a block with 77 points of blast resistance, but not a block with 78.
Explosion damage appears to not be additive (though the deobfuscated code is unclear enough that I'm not 100% certain either way), in that explosion damage seems to be checked for each individual explosion in a separate run of the explosion code and there's no code to save partial blast damage for later explosions to add to, so simultaneous explosions don't add together. We can see this in evidence by causing many, many explosions on a massive scale, and observing that the bedrock appears to remain entirely intact:
Best Answer
Yes. You just need a lot more of it.
I got three or four large ones... Oops.