No, proficiency bonuses never stack
Your proficiency bonus can’t be added to a single die
roll or other number more than once. - D&D Basic Rules V0.1 Chapter 1, P. 7
You can only ever receive a proficiency bonus once regardless of how many sources grant you proficiency in it. When picking skill and tool proficiencies granted by a background or class, Players should try to avoid overlapping proficiencies as the benefits are nil
However a proficiency bonus may be increased or decreased based on circumstances
Occasionally, your
proficiency bonus might be modified (doubled or halved,
for example) before you apply it. If a circumstance
suggests that your proficiency bonus applies more than
once to the same roll or that it should be multiplied
more than once, you nevertheless add it only once,
multiply it only once, and halve it only once. - D&D Basic Rules V0.1 Chapter 1, P. 7
DMs can use this as a way to give a bonus to skill checks based on good roleplay and/or planning.
The Rogue's Expertise feature is an exception
Expertise
At 1st level, choose two of your skill proficiencies, or
one of your skill proficiencies and your proficiency with
thieves’ tools. Your proficiency bonus is doubled for any
ability check you make that uses either of the chosen
proficiencies.
At 6th level, you can choose two more of your
proficiencies. - D&D Basic Rules V0.1 Chapter 3, P. 27
While not allowing proficiency bonuses to be counted more than once, expertise allows a Rogue to double the benefit of their proficiency bonus to two skills (or thieves' tools) at level 1 and again at level 6.
Backgrounds granting a redundant proficiency let you pick a different proficiency instead
In the Backgrounds section of the Basic Rules & PHB, the following rule is listed:
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
(Note that, while unclear, this rule may only apply to background proficiencies.)
Custom backgrounds are also an option available to players to avoid duplicate proficiencies due to backgrounds.
Other than the obvious Feat and multi-classing there is a section on training in the DMG and thus totally optional, but it is on p231.
- The character gains inspiration daily at dawn for 1d4+6 days.
- The character gains proficiency in a skill.
- The character gains a feat.
If you can get a feat by doing this or proficiency in a skill (pointing out that these don't count against your normal feat progression), I don't see anything game breaking to train for a single weapon. This is covered under unusual and usually unavailable training as a reward so it could be something to work toward by gaining renown with a guy that knows a guy for example.
This approach is entirely in the wheelhouse of DM allowance though and I would never allow it out of the box to be honest, as in Level 1.
This answer is based on the assumption that you already have a Wizard created and not necessarily an elf of any flavor, since the question was edited a few times by a couple of moderators this point may not have been clear, as in whether generating one from Level 1 or an existing Wizard of any race.
Best Answer
The only limit is the DM's approval
The optional rule from Tasha's states:
The is no limit imposed except that the rule itself is an optional rule (as stated at the start of the Character Options section) and therefore the DM may impose a limit to the number of swaps a player/character can make.