RelatedTo Query Using Both "and" and "or"
Crisis averted, thanks largely in part to the versitile nature of the relatedTo targetElement processing. The relevation came that I could feed an array of category ids (inclusive) into that element, and use a single relatedTo "and" parameter overall to get my intended filtering:

{% set params = {
    section: 'technologies',
    limit: null
} %}

{% set paramRelation = ['and'] %}
{% if technologyIds|length %}
    {% set paramRelation = paramRelation|merge([{ targetElement:technologyIds }]) %}
{% endif %}
{% if institutionId != "" %}
    {% set paramRelation = paramRelation|merge([{ targetElement:institutionId }]) %}
{% endif %}
{% if paramRelation|length &gt; 1 %}
    {% set params = params|merge({relatedTo: paramRelation}) %}
{% endif %}

{% set technologyResults = craft.entries(params) %}

{% paginate technologyResults.limit(10) as technologies %}
    ...
{% endpaginate %}


It's actually more simple this way as well, but I had initially underestimated how much heavy lifting the targetElement piece could do. Once again, Craft exceeds expectations. Thanks!